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Chapter 13: Q 21. (page 1066)

Set up the appropriate triple integral with spherical coordinates to show that
$M_{X Z} = \frac{1}{16} πk$

Short Answer

Expert verified

$M_{X Z} = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{3} \sin^{2} ϕ \sin θ d ρ d θ d ϕ$

Step by step solution

01

Given Information

We need to setup the an expression equivalent to $M_{X Z} = \frac{1}{16} πk$ involving spherical coordinates.

02

Simplification

The appropriate triple integral with spherical coordinates in this case will be

$M_{X Z} = \int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} k ρ^{3} \sin^{2} ϕ \sin θ d ρ d θ d ϕ$

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Most popular questions from this chapter

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

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

Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

$∭_{Ω} z ρ (x, y, z) d V$

Use the results of Exercises 59 and 60 to find the centers of masses of the laminæ in Exercises 61–67.

Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral $\int_{c}^{d} \int_{a}^{b} f (x, y) d x d y$ .

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$\int \int_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 \leq x \leq 4 a n d 0 \leq y \leq 3}$

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