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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)?
Answer
Outline
Outlines for
The Trapezoid Rule
Area Between Curves
Textbook
The Trapezoid Rule and Simpson's Rule
Area Between Curves
Today's Homework
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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 5subinterval trapezoid approximation of the integral from 0 to 9 of x^{3} + 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 y^{2} = 2x and x – y = 4. Sketch the region.
 Find the area between y = x^{2}/4 and y = x/2 + 2. Sketch the region.
 Consider the region between the circles x^{2} + y^{2} = a^{2} and x^{2} + y^{2} = b^{2} in the first quadrant. Divide this region into two pieces with the curve defined by x^{2}/a^{2} + y^{2}/b^{2} = 1 in the first quadrant. Find the ratio of the two regions created and sketch them.
Applets
 Numerical Integration
Videos

Estimate the integral of x^{2} dx from 0 to 6 using the Trapezoid Rule with 6 trapezoids.

Find the area between x^{2} and √x

Find the area between y = x^{2} and y = 2 – x^{2}

Find the area between the ellipse x^{2}/9 + y^{2} = 1 and the circle x^{2} + y^{2} = 1
