Chapter 13: Q 47 (page 1004)

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
role="math" localid="1650327788023" $\int \int_{R} y \sin x d A,$
whererole="math" localid="1650327080219" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

Short Answer

Expert verified

The value of double integral is :-

$\int \int_{R} y \sin x d A = \frac{1}{2}$ ,

where $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

Step by step solution

Step 1. Given Information

We have given the following double integral :-

$\int \int_{R} y \sin x d A,$

where $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

We have to evaluate this double integral

Step 2. Use iterated integrals

The given double integral is :-

$\int \int_{R} y \sin x d A,$

where $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$ .

Then by using Fubini's Theorem, we can writ this double integral as following :-

$\int \int_{R} y \sin x d A = \int_{0}^{1} \int_{0}^{\frac{π}{2}} y \sin x d x d y$ .

Then by using iterated integrals we have :-

$\int_{0}^{1} \int_{0}^{\frac{π}{2}} y \sin x d x d y = \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y$

Now we can solve this integral as following :-

$\int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \int_{0}^{1} (y \int_{0}^{\frac{π}{2}} \sin x d x) d y \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \int_{0}^{1} (y {[- \cos x]}^{\frac{π}{2}}_{0}) d y \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \int_{0}^{1} y [- \cos \frac{π}{2} - (- \cos 0)] d y \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \int_{0}^{1} y (0 + 1) d y \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \int_{0}^{1} y d y \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = {[\frac{y^{2}}{2}]}^{1}_{0} \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = [\frac{1}{2} - 0] \Rightarrow \int_{0}^{1} (\int_{0}^{\frac{π}{2}} y \sin x d x) d y = \frac{1}{2}$

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Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
role="math" localid="1650327788023" $\int \int_{R} y \sin x d A,$
whererole="math" localid="1650327080219" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use iterated integrals

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Evaluate each of the double integrals in Exercises 37-54as iterated integrals.role="math" localid="1650327788023" ∫∫RysinxdA,whererole="math" localid="1650327080219" R=x,y|0≤x≤π2and0≤y≤1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use iterated integrals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Logic and Functions

Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
role="math" localid="1650327788023" $\int \int_{R} y \sin x d A,$
whererole="math" localid="1650327080219" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$