Q6E Sketch the region whose area is ... [FREE SOLUTION]

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Sketch the region whose area is given by the integral and evaluate the integral.

\(\int\limits_{{\rm{\pi /2}}}^{\rm{\pi }} {\int\limits_{\rm{0}}^{{\rm{2 sin\theta }}} {{\rm{r dr d\theta }}} } \)

\(\int\limits_{\pi /2}^\pi {\int\limits_0^{2\sin \theta } {r{\rm{ }}dr{\rm{ }}d\theta } } = \frac{\pi }{2}\)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Introduction.

The given integral is,

\(I = \int\limits_{\pi /2}^\pi {\int\limits_0^{2\sin \theta } {r{\rm{ }}dr{\rm{ }}d\theta } } \)

Step 2: Evaluate the integral.

Integrating first on \(r\), we get

\(\begin{array}{l}I = \int\limits_{\pi /2}^\pi {\frac{1}{2}} \left. {{r^2}} \right|_0^{2\sin \theta }d\theta \\ = \int\limits_{\pi /2}^\pi {\frac{1}{2}} \left( {{2^2}{{\sin }^2}\theta - {0^1}} \right)d\theta \\ = \int\limits_{\pi /2}^\pi {2{{\sin }^2}\theta } {\rm{ }}d\theta \\{\sin ^2}\theta = \frac{{1 - \cos \left( {2\theta } \right)}}{2}\\\int\limits_{\pi /2}^\pi {2{{\sin }^2}\theta } {\rm{ }}d\theta = \int\limits_{\pi /2}^\pi {1 - \cos \left( {2\theta } \right)} {\rm{ }}d\theta \end{array}\)

Integrating on \(\theta \), we get

\(\begin{array}{l}I = \theta - \frac{1}{2}\sin \left. {2\theta } \right|_{\pi /2}^\pi \\ = \left( {\pi - 0} \right) - \left( {\frac{\pi }{2} - 0} \right)\\ = \pi - \frac{\pi }{2}\\ = \frac{\pi }{2}\end{array}\)

Step 3: Sketch the region.

The region R whose area is given by \(I = \int\limits_{\pi /2}^\pi {\int\limits_0^{2\sin \theta } {r{\rm{ }}dr{\rm{ }}d\theta } } \) is

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Essential Calculus: Early Transcendentals

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Short Answer

Sketch the region whose area is given by the integral and evaluate the integral.

Step by Step Solution

Step 1: Introduction.

Step 2: Evaluate the integral.

Step 3: Sketch the region.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Sketch the region whose area is given by the integral and evaluate the integral.

Step by Step Solution

Step 1: Introduction.

Step 2: Evaluate the integral.

Step 3: Sketch the region.

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