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Chapter 13: Q. 61 (page 1005)

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = \sin x \cos y W h e r e R = {(x, y) | 0 \leq x \leq π, 0 \leq y \leq π}$

Short Answer

Expert verified

The value of the integral is 0

Step by step solution

01

Given information

We are given a region

$f (x, y) = \sin x \cos y W h e r e R = {(x, y) | 0 \leq x \leq π, 0 \leq y \leq π}$

02

Find the volume

We are given

$\int_{R} \sin x \cos y = \int_{0}^{π} \int_{0}^{π} \sin x \cos y d x d y = [- \cos x \sin y] = 0$

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Most popular questions from this chapter

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

$\int \int_{R} x y^{3} d A W h e r e R = {(x, y) | - 2 \leq x \leq 2, - 1 \leq y \leq 1}$

In Exercises 61–64, let R be the rectangular solid defined by

$R = \{(x, y, z) | 0 \leq a_{1} \leq x \leq a_{2}, 0 \leq b_{1} \leq y \leq b_{2}, 0 \leq c_{2} \leq z \leq c_{2}\} .$

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2}) .$

(b) Verify that $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2})$ is the center of mass by using the appropriate integral expressions.

Evaluate the iterated integral :

$\int_{1}^{3} \int_{- 1}^{5} \int_{0}^{3} x y^{2} z d z d y d x$

Evaluate the sums in Exercises $23 - 28$ .

$\sum_{i = 1}^{4} \sum_{j = 1}^{3} \sum_{k = 1}^{2} i j^{2} k^{3}$

Evaluate the sums in Exercises $23 - 28$

$\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{m} i j^{2} k^{3}$

See all solutions

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