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

## Resources

Math 3 Course Syllabus
Practice Exams

# Contents

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)?

### Outline

Outlines for
The Trapezoid Rule
Area Between Curves

### Textbook

The Trapezoid Rule and Simpson's Rule
Area Between Curves

### Quiz

The Trapezoid Rule Quiz
Area Between Curves Quiz

### Examples

• 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.

### Applets

• Numerical Integration

### Videos

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

Lecture 23 | Index | Lecture 25