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<h1 class="center">Integral Approximations</h1>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/integral-area0.gif" alt="integral area" height="171" width="195"></p>
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<p> </p>
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<p><a href="integration-introduction.html">Integration</a> is the best way to find the <b>area</b> from a curve to the axis, because we get a formula for an <b>exact</b> answer.</p>
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<div style="clear:both"></div>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-area1.gif" alt="integral area small delta x" height="171" width="195"></p>
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<p class="right">But when integration is hard (or impossible) we can instead add up lots of slices to get an <b>approximate</b> answer.</p>
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<p>Let's have a go!</p>
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<h2>Examples</h2>
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<p>Let's use <b>f(x) = ln(x)</b> from x = 1 to x = 4</p>
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<p>We actually <i><b>can</b></i> integrate that (this let's us check answers) and get the true answer of <b>2.54517744447956....</b></p>
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<p>But <b> imagine we can't</b>, and all we can do is calculate values of ln(x):</p>
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<ul>
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<li>at x=1: ln(1) = 0</li>
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<li>at x=2: ln(2) = 0.693147...</li>
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<li>etc</li>
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</ul>
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<p>We will use a slice width of <b>1</b> to make it easy to see what is going on, but smaller slices are more accurate.</p>
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<p>There are different methods we can use:</p>
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<p> </p>
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<h3>Left Rectangular Approximation Method (LRAM)</h3>
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<p class="center"><img src="images/integral-approx-1.gif" alt="integral approximation Left Rectangular Graph" height="121" width="300"></p>
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<p>This method uses rectangles whose height is the left-most value. Areas are:</p>
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<ul>
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<li><i>x=1 to 2:</i> ln(1) × 1 = 0 × 1 = 0</li>
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<li><i>x=2 to 3:</i> ln(2) × 1 = 0.693147... × 1 = 0.693147...</li>
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<li><i>x=3 to 4:</i> ln(3) × 1 = 1.098612... × 1 = 1.098612...</li>
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</ul>
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<p>Adding these up gets <b>1.791759</b>, much lower than 2.545177. Why?</p>
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<p>Because we are missing all that area between the tops of the rectangles and the curve.</p>
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<p>This is made worse by a curve that is constantly increasing. When a curve goes up and down more, the error is usually less.</p>
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<h3> </h3>
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<h3>Right Rectangular Approximation Method (RRAM)</h3>
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<p class="center"><img src="images/integral-approx-2.gif" alt="integral approximation Right Rectangular Graph" height="124" width="300"></p>
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<p>Here we calculate the rectangle's height using the right-most value. Areas are:</p>
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<ul>
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<li><i>x=1 to 2:</i> ln(2) × 1 = 0.693147... × 1 = 0.693147...</li>
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<li><i>x=2 to 3:</i> ln(3) × 1 = 1.098612... × 1 = 1.098612...</li>
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<li><i>x=3 to 4:</i> ln(4) × 1 = 1.386294... × 1 = 1.386294...</li>
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</ul>
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<p>Adding these up gets <b>3.178054</b>, which is now much higher than 2.545177, because of the extra areas between the tops of the rectangles and the curve.</p>
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<h3> </h3>
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<h3>Midpoint Rectangular Approximation Method (MRAM)</h3>
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<p class="center"><img src="images/integral-approx-3.gif" alt="integral approximation Midpoint Rectangular Graph" height="121" width="300"></p>
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<p>We can also use the midpoint! Areas are:</p>
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<ul>
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<li><i>x=1 to 2:</i> ln(1.5) × 1 = 0.405465... × 1 = 0.405465...</li>
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<li><i>x=2 to 3:</i> ln(2.5) × 1 = 0.916291... × 1 = 0.916291...</li>
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<li><i>x=3 to 4:</i> ln(3.5) × 1 = 1.252763... × 1 = 1.252763...</li>
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</ul>
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<p>Adding these up gets <b>2.574519...</b>, which is quite close to 2.545177.</p>
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<p> </p>
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<h3>Trapezoidal Rule</h3>
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<p class="center"><img src="images/integral-approx-4.gif" alt="integral approximation: Trapezoidal Rule" height="118" width="300"></p>
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<p>We can have a sloped top! Each slice is now a <a href="../geometry/trapezoid.html">trapezoid</a> (or possibly a triangle), so it is called the Trapezoidal Rule.</p>
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<p class="center"><img src="images/integral-approx-6.gif" alt="integral approximation: Trapezoidal Rule zoomed in" height="130" width="300"></p>
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<p>The calculation just averages the left and right values. Areas are:</p>
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<ul>
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<li><i>x=1 to 2:</i> <span class="intbl">
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<em>ln(1) + ln(2)</em>
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<strong>2</strong>
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</span> × 1 = <span class="intbl">
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<em>0 + 0.693147...</em><strong>2</strong>
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</span> × 1 = 0.346573...</li>
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<li><i>x=2 to 3:</i> <span class="intbl">
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<em>ln(2) + ln(3)</em><strong>2</strong>
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</span> × 1 = <span class="intbl">
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<em>0.693147... + 1.098612...</em><strong>2</strong>
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</span> × 1 = 0.895879...</li>
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<li><i>x=3 to 4:</i> <span class="intbl"><em>ln(3) + ln(4)</em><strong>2</strong>
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</span> × 1 = <span class="intbl">
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<em>1.098612... + 1.386294...</em><strong>2</strong>
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</span> × 1 = 1.242453...</li>
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</ul>
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<p>Adding these up gets <b>2.484907</b>, which is still a bit lower than 2.545177, mostly because the curve is <a href="concave-up-down-convex.html">concave down</a> over the interval.</p>
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<p>Notice that in practice each value gets used twice (except first and last) and then the whole sum is divided by 2:</p>
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<p class="so"><span class="intbl">
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<em>ln(1) + ln(2)</em>
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<strong>2</strong>
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</span> × 1 + <span class="intbl">
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<em>ln(2) + ln(3)</em>
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<strong>2</strong>
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</span> × 1 + <span class="intbl">
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<em>ln(3) + ln(4)</em>
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<strong>2</strong>
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</span> × 1</p>
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<p class="so"><span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span> × ( ln(1) + ln(2) + ln(2) + ln(3) + ln(3) + ln(4) )</p>
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<p class="so"><span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span> × ( ln(1) + 2 ln(2) + 2 ln(3) + ln(4) )</p>
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<p>So we can have a general formula:</p>
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<p class="so"><span class="intbl">
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<em>Δx</em>
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<strong>2</strong>
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</span> × ( f(x<sub>0</sub>) + 2f(x<sub>1</sub>) + 2f(x<sub>2</sub>) + ... 2f(x<sub>n−1</sub>) + f(x<sub>n</sub>) )</p>
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<p>By the way, this method is just the average of the Left and Right Methods:</p>
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<p class="center">Trapezoidal Approximation = <span class="intbl">
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<em>LRAM + RRAM</em>
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<strong>2</strong>
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</span></p><p> </p>
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<p class="words">Note: the previous 4 methods are also called <b>Riemann Sums</b> after the mathematician Bernhard Riemann.</p>
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<p> </p>
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<h3>Simpson's Rule</h3>
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<p class="center"><img src="images/integral-approx-5.gif" alt="integral approximation: Simpsons Rule" height="118" width="300"></p>
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<p>An improvement on the Trapezoidal Rule is Simpson's Rule. It is based on using parabolas at the top instead of straight lines. The parabolas often get quite close to the real curve:</p>
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<p class="center"><img src="images/integral-approx-7.gif" alt="integral approximation: Simpsons Rule zoomed in" height="194" width="300"></p>
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<p>It sounds hard, but we end up with a formula like the trapezoid formula, but we divide by 3 and use a 1, 4, 2, ..., 2, 4, 1 pattern of factors:</p>
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<p class="so"><span class="intbl">
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<em>Δx</em>
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<strong>3</strong>
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</span> × ( f(x<sub>0</sub>) + 4f(x<sub>1</sub>) + 2f(x<sub>2</sub>) + ... 4f(x<sub>n−1</sub>) + f(x<sub>n</sub>) )</p>
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<p>But: <b>n must be even</b>. So let's take 6 slices of 0.5 each:</p>
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<p class="so"><span class="intbl">
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<em>0.5</em>
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<strong>3</strong>
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</span> × ( f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + 2f(3) + 4f(3.5) + f(4) )</p>
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<p>Plugging in values of ln(1) etc gives:</p>
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<p class="so"><span class="intbl">
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<em>0.5</em>
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<strong>3</strong>
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</span> × ( 15.2679... )</p>
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<p class="so">2.544648...</p>
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<p>This is a great result when compared to <b>2.545177...</b>.</p>
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<h2>Plus and Minus</h2>
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<p><span style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-def-cos-1-3.svg" alt="definite integral cos(x) from 1 to 3" height="127" width="212"></span></p>
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<p>When the curve is below the axis the value of the integral is negative!</p>
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<p><b>So we get a "net" value.</b></p>
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<div class="center80">If we want a total <b>area</b> (say we wanted to paint it) we can use the <a href="../sets/function-absolute-value.html">absolute value</a> function <b>abs()</b>. Or manually find where the curve crosses the axis and then work out separate integrals and reverse the negatives before adding.</div>
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<h2>Error and Accuracy</h2>
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<p>Let's compare them all:</p>
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<div class="simple">
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<table align="center" cellpadding="5">
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<tbody>
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<tr>
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<td class="large"><b>f(x)=ln(x)</b></td>
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<td style="text-align:center;">N = 3</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">N = 6</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">N = 100</td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:center;">Estimation</td>
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<td style="text-align:center;"><i>Error</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">Estimation</td>
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<td style="text-align:center;"><i>Error</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">Estimation</td>
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<td style="text-align:center;"><i>Error</i></td>
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</tr>
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<tr>
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<td><b>LRAM</b></td>
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<td style="text-align:center;">1.791759</td>
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<td style="text-align:center;"><i>0.753418</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.183140</td>
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<td style="text-align:center;"><i>0.362037</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.524327</td>
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<td style="text-align:center;"><i>0.020850</i></td>
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</tr>
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<tr>
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<td><b>RRAM</b> </td>
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<td style="text-align:center;">3.178054</td>
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<td style="text-align:center;"><i>-0.632877</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.876287</td>
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<td style="text-align:center;"><i>-0.331110</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.565916</td>
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<td style="text-align:center;"><i>-0.020739</i></td>
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</tr>
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<tr>
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<td><b>MRAM</b> </td>
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<td style="text-align:center;">2.574519</td>
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<td style="text-align:center;"><i>-0.029342</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.552851</td>
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<td style="text-align:center;"><i>-0.007674</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.545206</td>
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<td style="text-align:center;"><i>-0.000029</i></td>
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</tr>
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<tr>
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<td><b>Trapezoidal Rule</b></td>
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<td style="text-align:center;">2.484907</td>
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<td style="text-align:center;"><i>0.060271</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.529713</td>
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<td style="text-align:center;"><i>0.015464</i></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">2.545121</td>
|
||
<td style="text-align:center;"><i>0.000055</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td><b>Simpson’s Rule</b></td>
|
||
<td colspan="2" align="center"><i>(N must be even)</i></td>
|
||
<td style="text-align:center;"> </td>
|
||
<td style="text-align:center;">2.544648</td>
|
||
<td style="text-align:center;"><i>0.000529</i></td>
|
||
<td style="text-align:center;"> </td>
|
||
<td style="text-align:center;">2.545177</td>
|
||
<td style="text-align:center;"><i><0.000001</i></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p class="center"><b>Simpson’s Rule</b> rules! And it is just as easy to use as the others.</p>
|
||
<p>Of course a different function will produce different results. Why not try some yourself?</p>
|
||
|
||
|
||
<h2>Maximum Error</h2>
|
||
|
||
<p>Often we won't know the actual answer ... so how do we know how good our estimate is?</p>
|
||
<p>You can get a good feel by trying different slice widths.</p>
|
||
<p>And there are also these <b>formulas for the maximum error</b> of approximation (these are for the worst case and the actual error will hopefully be a lot smaller):</p>
|
||
<p class="center large">For Midpoint: |E| =
|
||
<span class="intbl">
|
||
<em>K(b-a)<sup>3</sup></em>
|
||
<strong>24n<sup>2</sup></strong>
|
||
</span></p>
|
||
<p class="center large">For Trapezoidal: |E| =
|
||
<span class="intbl">
|
||
<em>K(b-a)<sup>3</sup></em>
|
||
<strong>12n<sup>2</sup></strong>
|
||
</span></p>
|
||
<p class="center large">For Simpson's: |E| =
|
||
<span class="intbl">
|
||
<em>M(b-a)<sup>5</sup></em>
|
||
<strong>180n<sup>4</sup></strong>
|
||
</span></p>
|
||
<p>Where:</p>
|
||
<ul>
|
||
<li><b>|E|</b> is the <a href="../numbers/absolute-value.html">absolute value</a> of the maximum error (could be plus or minus)</li>
|
||
<li><b>a</b> is the start of the interval</li>
|
||
<li><b>b</b> is the end of the interval</li>
|
||
<li><b>n</b> is the number of slices</li>
|
||
<li><b>K</b> is the greatest second derivative over the interval.</li>
|
||
<li><b>M</b> is the greatest fourth derivative over the interval.</li>
|
||
</ul>
|
||
<p>(By "greatest" we mean the maximum absolute value.)</p>
|
||
<p><b>a</b>, <b>b</b> and <b>n</b> are easy, but how do we find <b>K</b> and <b>M</b> ?</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: f(x) = ln(x) between 1 and 4</h3>
|
||
<p>Let's find some derivatives first, we will need them:</p>
|
||
<ul>
|
||
<li>1<sup>st</sup> derivative: f'(x) = 1/x</li>
|
||
<li>2<sup>nd</sup> derivative: f''(x) = −1/x<sup>2</sup></li>
|
||
<li>3<sup>rd</sup> derivative: f<sup>(3)</sup>(x) = 2/x<sup>3</sup></li>
|
||
<li>4<sup>th</sup> derivative: f<sup>(4)</sup>(x) = −6/x<sup>4</sup></li>
|
||
<li>5<sup>th</sup> derivative: f<sup>(5)</sup>(x) = 24/x<sup>5</sup></li>
|
||
</ul>
|
||
<p>The greatest K could be at the start, end, or somewhere in between:</p>
|
||
<ul>
|
||
<li>Start: f''(1) = −1/1<sup>2</sup> = −1</li>
|
||
<li>End: f''(4) = −1/4<sup>2</sup> = −1/16</li>
|
||
<li>In between: use the 3<sup>rd</sup> derivative to see if there are zeros in the 1 to 4 interval, which could mean a change in direction.<br>
|
||
Does f<sup>(3)</sup>(x) = 0 between 1 and 4? No. So the maximum is at start or end.</li>
|
||
</ul>
|
||
<p>So K = 1 (the maximum absolute value)</p>
|
||
<p>Same for M, but higher derivatives:</p>
|
||
<ul>
|
||
<li>Start: f<sup>(4)</sup>(1) = −6/1<sup>4</sup> = −6</li>
|
||
<li>End: f<sup>(4)</sup>(4) = −6/4<sup>4</sup>= −6/256</li>
|
||
<li>In between: use the 5<sup>th</sup> derivative to see if there are zeros in the 1 to 4 interval.<br>
|
||
Does f<sup>(5)</sup>(x) = 24/x<sup>5</sup> equal zero between 1 and 4? No.</li>
|
||
</ul>
|
||
<p>So M = 6 (the maximum absolute value)</p>
|
||
<p>For just 6 slices, the Maximum Errors are:</p>
|
||
<p class="center large">Midpoint: |E| = <span class="intbl">
|
||
<em>1(4−1)<sup>3</sup></em>
|
||
<strong>24×6<sup>2</sup></strong>
|
||
</span> = 0.03125</p>
|
||
<p class="center large">Trapezoidal: |E| = <span class="intbl">
|
||
<em>1(4−1)<sup>3</sup></em>
|
||
<strong>12×6<sup>2</sup></strong>
|
||
</span> = 0.0625</p>
|
||
<p class="center large">Simpson's: |E| = <span class="intbl">
|
||
<em>6(4−1)<sup>5</sup></em>
|
||
<strong>180×6<sup>4</sup></strong>
|
||
</span> = 0.00625</p>
|
||
</div>
|
||
|
||
|
||
<h2>Shapes we Know</h2>
|
||
|
||
<p>The curve may have a shape we know, and we can use geometry formulas like these examples:</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: Triangle</h3>
|
||
<p class="center"><img src="images/integral-approx-tri.gif" alt="integral approximation triangle" height="178" width="182"><br>
|
||
<b>f(x) = 2 − x</b>, from 0 to 2</p>
|
||
<p class="center larger">A = ½ × 2 × 2 = <b>2</b></p>
|
||
</div>
|
||
<div class="example">
|
||
|
||
<h3>Example: Rectangle</h3>
|
||
<p class="center"><img src="images/integral-approx-rect.gif" alt="integral approximation rectangle" height="132" width="173"><br>
|
||
<b>f(x) = 2</b>, from 0 to 3</p>
|
||
<p class="center larger">A = 2 × 3 = <b>6</b></p>
|
||
</div>
|
||
<div class="example">
|
||
|
||
<h3>Example: Semicircle</h3>
|
||
<p class="center"><img src="images/integral-approx-circ.gif" alt="integral approximation circle" height="96" width="289"><br>
|
||
<b>f(x) = √(1 − x<sup>2</sup>)</b>, from −1 to +1</p>
|
||
<p class="center larger">A = <span class="times">π</span> r<sup>2</sup> / 2 = <span class="times">π</span> / 2</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Conclusion</h2>
|
||
|
||
<p class="dotpoint">We can estimate the area under a curve by slicing a function up</p>
|
||
<div class="dotpoint">There are many ways of finding the area of each slice such as:
|
||
|
||
|
||
<ul>
|
||
<li>Left Rectangular Approximation Method (LRAM)</li>
|
||
<li>Right Rectangular Approximation Method (RRAM)</li>
|
||
<li>Midpoint Rectangular Approximation Method (MRAM)</li>
|
||
<li>Trapezoidal Rule</li>
|
||
<li>Simpson's Rule</li>
|
||
</ul>
|
||
</div>
|
||
<div class="dotpoint">We can use error formulas to find the largest possible error in our estimate
|
||
</div>
|
||
<div class="dotpoint">Basic geometry formulas can sometimes help us find areas under the curve</div>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="integration-introduction.html">Integration</a>
|
||
<a href="integral-approximation-calculator.html">Integral Approximations Calculator and Graph</a>
|
||
<a href="derivative-vs-integral.html">Graphical Intro to Derivatives and Integrals</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
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