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Chapter 13: Q. 39 (page 1015)

In Exercises 35鈥�40, find the volume of the solid bounded above by the given function over the specified region
$f (x, y) = \sin x \cos y$ and $惟 = {(x, y) | 鈭� 2 鈮� x 鈮� 2$ and $| x | 鈮� y 鈮� 2}$

Short Answer

Expert verified

The volume of the described solid is: 0

Step by step solution

01

Step 1. Given Information

A function, $f (x, y) = \sin x \cos y$

The region, $惟 = {(x, y) | 鈭� 2 鈮� x 鈮� 2$ and $| x | 鈮� y 鈮� 2}$

02

Step 2. Finding the volume of given solid

The double integral to find the volume of the given solid is given by,

{鈭�}_{| x |}^{2} {鈭�}_{- 2}^{2} (\sin x) (\cos y) d x d y = {鈭�}_{| x |}^{2} {鈭�}_{- 2}^{2} (\sin x) (\cos y) d x d y = {鈭�}_{| x |}^{2} (\cos y) [{(- \cos x)}_{- 2}^{2}] d y = {鈭�}_{| x |}^{2} (\cos y) [- \cos (2) + c o s (- 2)] d y = {鈭�}_{| x |}^{2} (\cos y) [- \cos (2) + c o s (2)] d y = {鈭�}_{| x |}^{2} (\cos y) [0] d y = 0

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral:

${鈭�}_{鈭� 2}^{4} 鈥� {鈭�}_{2}^{6} 鈥� {鈭�}_{0}^{5} 鈥� f (x, y, z) d y d x d z$

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" $惟_{1}, 惟_{2}, 鈥�, 惟_{n} .$ Show that if the masses of these lamin忙 are $m_{1}, m_{2}, 鈥�, m_{n}$ and the centers of masses are $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}), ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2}), 鈥�, ({\overset{炉}{x}}_{n}, {\overset{炉}{y}}_{n}),$ then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{x}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} and \overset{炉}{y} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{y}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} .$

$How many summands are in {鈭�}_{j = 3}^{13} {鈭�}_{k = 5}^{20} \frac{j^{2}}{e^{jk}} ?$

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

Find the masses of the solids described in Exercises 53鈥�56.

The first-octant solid bounded by the coordinate planes and the plane 3x + 4y + 6z = 12 if the density at each point is proportional to the distance of the point from the xz-plane.

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