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Chapter 5: Q. 1 TB (page 416)
Expressing geometric quantities with integrals: Express each of the given geometric quantities in terms of definite integrals. You do not have to solve the integrals.
The signed area of the region between the graph of and the x-axis on
The integral will be
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Explain why and are essentially the same integral after a change of variables.
Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?
Solve the following two ways:
(a) with the trigonometric substitution x = 3 tan u;
(b) with algebra and the derivative of the arctangent.
Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
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