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Use a CAS double-integral evaluator to find the integrals. Then reverse the order of integration and evaluate, again with a CAS.$$\int_{0}^{3} \int_{x^{2}}^{9} x \cos \left(y^{2}\right) d y d x$$

$y=9$,$y=x^{\wedge} 2$

Calculus 3

Chapter 15

Multiple Integrals

Section 2

Double Integrals over General Regions

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Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

03:17

Use a CAS double-integral …

03:32

00:49

02:50

03:50

the given double intra Galis 0 to 3. X square to nine. Ex Hussein y square Deal I d. X on. We have to evaluate this interval by using the graphing calculator. So we'll use our graphing calculator and well, get the value off this integral US negative 0.1 Fife 747 and no, we have to write this integral by reversing the order of integration. And then again, we have to user graphing calculator to estimate the value off the integral. So first of all, I'll going to reverse the order of integration. For that, I have to sketch the region of integration on for this. We know that these are the limits for why that means why is equals Toe X Square. Unwise equals to nine on these limits for X net meant X is equal to zero. X is equal to three. Now I'll be taking graph paper and we're goingto graph these equation or a graph paper so that we can sketch the reason off integration. So this is the Y X is on. This is X axis on this golf represents. Why is equal to X square on this line represents X is it cools to three on this line. Represents why is equals to nine on Dhere X is equal to zero Now the reason off integration will be this region. We have to but it right just integral by reversing the order of integration. So you get a doubling trickle. Excuse ain y square, dx, dy y and we have to take the horizontal limit the horizontal cross section to find the limit of this integral. So here, x zero that mean the limit will be zero Hair X is equal. It's too. We know that the goal of this equation is vice vehicles two x squared from her ex will be close to Underwood. Why so under roadway on the limits for divide will be wearing between they slain to the slain. So here, why is zero on Dhere? Why is in night now again, we'll use a graphing calculator to evaluate or you could say, to estimate the value of this integral and it comes out to be as negative 0.1574 soon. So finally we can conclude that the devil in Tegel 0 to 3 x squared to nine. Excuse ain y square de Vie de X is equal to 0 to 9 Integral, integral zero to root way Excuse ain Vice Square be X Do y on it is equals to negative 0.15747 So this is a finer reserved

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