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Q5E

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Essential Calculus: Early Transcendentals

Found in: Page 707

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Evaluate the iterated integral \(\int_0^1 {\int_0^{{s^2}} {cos} } \left( {{s^3}} \right)dtds\)

Value of integral is \(\frac{1}{3}sin(1)\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Integration on \(t\).

\(\int_0^1 {\int_0^{{s^2}} {cos} } \left( {{s^3}} \right)dtds = \left. {\int_0^1 t cos\left( {{s^3}} \right)} \right|_0^{{s^2}}ds\)

\( = \int_0^1 {{s^2}} cos\left( {{s^3}} \right)ds\)

Step 2: Integrate on \(S\).

Using the substitution

\(u = {s^3}\) and \(\)

\(du = 3{s^2}\,ds\)

\(\int_0^1 {{s^2}} cos\left( {{s^3}} \right)ds = \int_0^1 {\frac{1}{3}} cos(u)du\)

\( = \left. {\frac{1}{3}sin(u)} \right|_0^1 = \frac{1}{3}sin(1)\)

Hence, value of integral is \(\frac{1}{3}sin(1)\)

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