Chapter 13: Q. 29 (page 1055)

Evaluate the iterated integral :
$\int_{0}^{π / 2} \int_{0}^{\cos y} \int_{0}^{\sin y} (2 x + y) d z d x d y$

Short Answer

Expert verified

$\frac{π}{8} + \frac{1}{3}$

Step by step solution

Step 1. Given information.

Given integral is :

$\int_{0}^{π / 2} \int_{0}^{\cos y} \int_{0}^{\sin y} (2 x + y) d z d x d y$

We have to evaluate it.

Step 2. Evaluate.

Given

$\int_{0}^{π / 2} \int_{0}^{\cos y} \int_{0}^{\sin y} (2 x + y) d z d x d y$

$\begin{aligned} = \int_{0}^{\frac{π}{2}} \int_{0}^{\cos y} [\int_{0}^{\sin y} (2 x + y) d z] d x d y \\ = \int_{0}^{\frac{π}{2}} \int_{0}^{\cos y} [2 x \int_{0}^{\sin y} d z + y \int_{0}^{\sin y} d z] d x d y \\ = \int_{0}^{\frac{π}{2}} \int_{0}^{\cos y} [2 x (z)_{0}^{\sin y} + y (z)_{0}^{\sin y}] d x d y \\ = \int_{0}^{\frac{π}{2}} \int_{0}^{\cos y} [2 x \sin y + y \sin y] d x d y \end{aligned}$

Step 3. Integrate with respect to x.

Integrate with respect to x

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

Integrate with respect to y

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

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Evaluate the iterated integral :
$\int_{0}^{π / 2} \int_{0}^{\cos y} \int_{0}^{\sin y} (2 x + y) d z d x d y$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Evaluate.

Step 3. Integrate with respect to x.

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Step by step solution

Step 1. Given information.

Step 2. Evaluate.

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