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Chapter 13: Q. 14 (page 1003)

Explain why using an iterated integral to evaluate a double integral is often easier than using the definition of the double integral to evaluate the integral.

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Most popular questions from this chapter

Earlier in this section, we showed that we could use Fubini’s theorem to evaluate the integral $\int \int_{R} x^{2} y d A$ and we showed that $\int_{2}^{5} \int_{1}^{3} x^{2} y d x d y = 91$ Now evaluate the double integral by evaluating the iterated integral $\int_{1}^{3} \int_{2}^{5} x^{2} y d y d x$ .

Explain how to construct a Riemann sum for a function of three variables over a rectangular solid.

In Exercises, let $S = \{(x, y) ∣ x^{2} + y^{2} \leq 1 and x \geq 0\} .$

If the density at each point in S is proportional to the point’s distance from the origin, find the moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of S about the x-axis, the y-axis, and the origin.

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{aligned} ∭_{R} \ln (x y z^{2}) d V, where \\ R = {(x, y, z) ∣ 1 \leq x \leq 3, 1 \leq y \leq e, and 1 \leq z \leq 2} \end{aligned}$

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

$\int \int_{R} x^{2} y d A W h e r e R = {(x, y) | - 1 \leq x \leq 0, 0 \leq y \leq 2}$

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Explain why using an iterated integral to evaluate a double integral is often easier than using the definition of the double integral to evaluate the integral.

Short Answer

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Step 1. Given information:

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