<p>Use the sketch to split the shaded region into curved and straight-edged pieces, then add or subtract their areas.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjIwMCwiaGVpZ2h0IjoxNTAsImVsZW1lbnRzIjpbeyJpZCI6InUyMC10NC1jMS1pMSIsInRvIjo2LCJmcm9tIjoyLCJ0eXBlIjoiaW50ZWdyYWwiLCJjb2xvciI6IiMyNTYzZWIiLCJmaWxsQ29sb3IiOiIjMjU2M2ViIiwiZXhwcmVzc2lvbiI6IjgqeC14XjIiLCJleHByZXNzaW9uMiI6IjEyIiwiZmlsbE9wYWNpdHkiOjAuMTh9LHsiaWQiOiJ1MjAtdDQtYzEtZjEiLCJ0eXBlIjoiZnVuY3Rpb25ncmFwaCIsImNvbG9yIjoiIzI1NjNlYiIsInhPZmZzZXQiOjAsInlPZmZzZXQiOjAsImV4cHJlc3Npb24iOiI4KngteF4yIiwic3Ryb2tlV2lkdGgiOjJ9LHsiaWQiOiJ1MjAtdDQtYzEtZjIiLCJ0eXBlIjoiZnVuY3Rpb25ncmFwaCIsImNvbG9yIjoiI2RjMjYyNiIsInhPZmZzZXQiOjAsInlPZmZzZXQiOjAsImV4cHJlc3Npb24iOiIxMiIsInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJ1MjAtdDQtYzEtc2VnMSIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjOWNhM2FmIiwicG9pbnQxIjpbMiwwXSwicG9pbnQyIjpbMiwxMl0sInN0cm9rZVdpZHRoIjoxfSx7ImlkIjoidTIwLXQ0LWMxLXNlZzIiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzljYTNhZiIsInBvaW50MSI6WzYsMF0sInBvaW50MiI6WzYsMTJdLCJzdHJva2VXaWR0aCI6MX0seyJ4IjoyLCJ5IjowLCJpZCI6InUyMC10NC1jMS1wMSIsInR5cGUiOiJwb2ludCIsImZpeGVkIjp0cnVlLCJsYWJlbCI6ImEiLCJsYWJlbE9mZnNldCI6Wy02LC0xNl19LHsieCI6NiwieSI6MCwiaWQiOiJ1MjAtdDQtYzEtcDIiLCJ0eXBlIjoicG9pbnQiLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJiIiwibGFiZWxPZmZzZXQiOlsyLC0xNl19XSwic2hvd1hBeGlzIjp0cnVlLCJzaG93WUF4aXMiOnRydWUsImJvdW5kaW5nQm94IjpbLTAuNiwxOCw4LjYsLTIuNV0sInNob3dBeGlzTGFiZWxzIjpmYWxzZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOmZhbHNlfQ==" data-width="200" data-height="150" data-proportional style="width:200px;height:150px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:200px;--graph-aspect:200 / 150;"></div><p>Horizontal line: \(A=\displaystyle\int_a^b y\,\mathrm{d}x-\text{area of rectangle}\)</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="200" data-height="150" data-proportional style="width:200px;height:150px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:200px;--graph-aspect:200 / 150;"></div><p>Sloping line: \(A=\displaystyle\int_a^b y\,\mathrm{d}x-\text{area of triangle (or trapezium)}\)</p></div></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 480px;"><colgroup><col style="width: 37.5000%;"><col style="width: 62.5000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Shape</strong></th><th style="text-align: center; --cell-justify: center"><strong>Area</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Rectangle</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Triangle</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Trapezium</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Integrating a straight-line boundary is equally valid. Use perpendicular heights in the geometric formulae.</p>

Use the sketch to split the shaded region into curved and straight-edged pieces, then add or subtract their areas.

\(\dfrac{1}{2}\times\text{base}\times\text{height}\)

Integrating a straight-line boundary is equally valid. Use perpendicular heights in the geometric formulae.

Combining Integrals with Rectangles, Triangles and Trapeziums

Combining Integrals with Rectangles, Triangles and Trapeziums

Shape	Area
Rectangle
Triangle
Trapezium