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Evaluate the given double integrals.$$\int_{0}^{\pi / 6} \int_{\pi / 3}^{y} \sin x d x d y$$

$\frac{\pi-6}{12}$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 4

Double Integrals

Partial Derivatives

Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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we want to evaluate the double integral given by the integral from zero to pi over six of the integral. From pi over three to Y. Our sin X. Dx Dy. So this question is found in our understanding of integration from the standpoint of Multivariate calculus are such let's proceed to solve. So to solve we need to understand how to take an integral. What we're going to do is you single variable integration techniques. We picked up in single variable calculus one by one where first we take the inner integral which in this case is the integral pi over three to Y of X. Dx. Once we evaluate that we plug it into the outer integral from zero to pi over 60 Y to solve. So we have in our double integral is equal to negative integral. Zero to pi over six. Cossacks from probably three dy dy or negative zero to pi over six integral coastline is happy. Why? Thus we can evaluate as follows, we have negative sign one minus one half white, from zero to pi over six, or negative one half minus pi over 12, which produces the final solution, pi minus 6/12.

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