Chapter 13: Q. 33 (page 1055)

Evaluate the triple integrals over the specified rectangular solid region.
$\begin{aligned} ∭_{R} z \sin x \cos y d V, where \\ R = \{(x, y, z) ∣ 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, and 1 \leq z \leq 3\} \end{aligned}$

Short Answer

Expert verified

$∭_{R} z \sin x \cos y d V = 8$

Step by step solution

Step 1. Given information.

We have been given the triple integral:

$\begin{array}{r} ∭_{R} z \sin x \cos y d V, where \\ R = \{(x, y, z) ∣ 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, and 1 \leq z \leq 3\} \end{array}$

We have to evaluate this over the specified rectangular solid regions.

Step 2. Evaluate.

By Fubini's theorem of triple integral :

$\begin{aligned} ∭_{R} z \sin x \cos y d V = \int_{0}^{π / 2 π} \int_{\frac{3 π}{2}}^{3} \int_{1}^{3} z \sin x \cos d x d y d z \\ = \int_{0}^{π} \int_{\frac{3 π}{2}}^{2 π} [\int_{1}^{3} z \sin x \cos y d z] d y d x \\ = \int_{0}^{π} \int_{\frac{3 π}{2}}^{2 π} [\sin x \cos y \int_{1}^{3} z d z] d y d x \\ = \int_{0}^{π} \int_{\frac{3 π}{2}}^{2 π} [\sin x \cos y {[\frac{z^{2}}{2}]}_{1}^{3}] d y d x \\ = \int_{0}^{π} \int_{\frac{3 π}{2}}^{2 π} [\sin x \cos y [\frac{3^{2}}{2} - \frac{1^{2}}{2}] d y d x \\ = \int_{0}^{π} \int_{\frac{3 π}{2}}^{2 π} [4 \sin x \cos y] d y d x \end{aligned}$

Step 3. Integrate with respect to y.

Integrate with respect to y

$\begin{aligned} = \int_{0}^{π} [4 \sin x \int_{\frac{3 π}{2}}^{2 π} [\cos y] d y] d x \\ = \int_{0}^{π} [4 \sin x [\sin y]_{\frac{3 π}{2}}^{2 π}] d x \\ = \int_{0}^{π} [4 \sin x [\sin 2 π - \sin \frac{3 π}{2}]] d x \\ = \int_{0}^{π} [4 \sin x [0 - (- 1)]] d x \\ = \int_{0}^{π} [4 \sin x] d x \\ = 4 [\int_{0}^{π} (\sin x)] \\ = 4 [(- \cos x)]_{0}^{π} \\ = - 4 [(\cos π) - (\cos 0)] \\ = - 4 [- 1 - 1] \\ = - 4 [- 2] \\ = 8 \end{aligned}$

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Evaluate the triple integrals over the specified rectangular solid region.
$\begin{aligned} ∭_{R} z \sin x \cos y d V, where \\ R = \{(x, y, z) ∣ 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, and 1 \leq z \leq 3\} \end{aligned}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to y.

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Evaluate the triple integrals over the specified rectangular solid region.∭R zsin⁡xcos⁡ydV,whereR=(x,y,z)∣0≤x≤π,3π2≤y≤2π,and1≤z≤3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to y.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Decision Maths

Probability and Statistics

Pure Maths

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Study anywhere. Anytime. Across all devices.

Evaluate the triple integrals over the specified rectangular solid region.
$\begin{aligned} ∭_{R} z \sin x \cos y d V, where \\ R = \{(x, y, z) ∣ 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, and 1 \leq z \leq 3\} \end{aligned}$