Calculus on Demand at Dartmouth College Lecture 23 | Index | Lecture 25
Lecture 24


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In this lecture we discuss the Trapezoid Rule, a numerical technique for evaluating (approximately) a definite integral. We also discuss a formula for finding the area between two curves.

Quick Question

What is the area between the two curves f(x) = 1 − x/2 (in red) and g(x) = x/2 (in blue)?



Outlines for
The Trapezoid Rule
Area Between Curves


The Trapezoid Rule and Simpson's Rule
Area Between Curves

Today's Homework

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The Trapezoid Rule Quiz
Area Between Curves Quiz


  • Approximate the integral from 0 to 2 of √x with 4 trapezoids. Sketch a figure showing the curve and the trapezoids involved. Compare your answer with the answer you find using integration formulas.
  • Compare the 5-subinterval trapezoid approximation of the integral from 0 to 9 of x3 + 3 with the exact value of the integral. How great is the difference between them?
  • How accurate is the Trapezoid Rule for approximating integrals?
  • Find the area of the region bounded by y2 = 2x and x – y = 4. Sketch the region.
  • Find the area between y = x2/4 and y = x/2 + 2. Sketch the region.
  • Consider the region between the circles x2 + y2 = a2 and x2 + y2 = b2 in the first quadrant. Divide this region into two pieces with the curve defined by x2/a2 + y2/b2 = 1 in the first quadrant. Find the ratio of the two regions created and sketch them.


  • Click to see the appletNumerical Integration


  • click to see the video Estimate the integral of x2 dx from 0 to 6 using the Trapezoid Rule with 6 trapezoids.
  • click to see the videoFind the area between x2 and √x
  • click to see the videoFind the area between y = x2 and y = 2 – x2
  • click to see the videoFind the area between the ellipse x2/9 + y2 = 1 and the circle x2 + y2 = 1

Lecture 23 | Index | Lecture 25