To evaluate the double integral, ∫π2π∫0x3sinyxdydx.…

Answered: 2T sin (2) dydx 0,

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.1: Equations

Problem 49E

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To evaluate the double integral, $\int_{\sqrt{π}}^{\sqrt{2 π}} \int_{0}^{x^{3}} \sin (\frac{y}{x}) d y d x$ .

$\begin{array}{rcl} \int_{\sqrt{π}}^{\sqrt{2 π}} \int_{0}^{x^{3}} \sin (\frac{y}{x}) d y d x & = & \int_{\sqrt{π}}^{\sqrt{2 π}} {[- x \cos (\frac{y}{x})]}_{0}^{x^{3}} d x \\ = & \int_{\sqrt{π}}^{\sqrt{2 π}} (- x \cos (x^{2}) + x) d x \\ = & \int_{\sqrt{π}}^{\sqrt{2 π}} (1 - \cos (x^{2})) x d x \end{array}$

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