Calculate: {eq}\int_{-3}^{3} \int_{0}^{\frac{\pi}{2}} \, (y + y^2 \cos x) \, \mathrm{d}x \, \mathrm{d}y {/eq}.


Calculate: {eq}\int_{-3}^{3} \int_{0}^{\frac{\pi}{2}} \, (y + y^2 \cos x) \, \mathrm{d}x \, \mathrm{d}y {/eq}.

Iterated Integrals:

We generally evaluate double integrals as iterated integrals. To integrate {eq}\int_a^b \int_c^d f(x,y)\, dy\, dx {/eq}, we first integrate {eq}f {/eq} assuming that {eq}x {/eq} is a constant and then evaluate the result with {eq}c {/eq} and {eq}d {/eq} for {eq}y {/eq}. Finally we integrate the result with respect to {eq}x {/eq}.

If the roles of the variables are reversed we integrate with respect to {eq}x {/eq} first and then {eq}y {/eq}.

Answer and Explanation: 1

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We calculate this integral inline

{eq}\begin{align} \int_{-3}^3 \int_0^{\frac{\pi}{2}} (y+y^2\cos x)\, dx\, dy&=\int_{-3}^3 (yx+y^2\sin...

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Double Integrals & Evaluation by Iterated Integrals


Chapter 15 / Lesson 4

Learn about iterated integrals and discover how to evaluate a double integral. Explore how iterative integration works and see how to solve double integrals.

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