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Determine the region $R$ determined by the given double integral.$$\int_{0}^{2} \int_{2 x^{2}}^{8} f(x, y) d y d x$$

Bounded by $y=2 x^{2}, y=8$ and $x=0,$ see Fig. Ex. 16

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Determine the region $R$ d…

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Evaluate the double integr…

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Find each double integral.…

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For this problem, we are asked to determine the region are determined by the given double integral from 0 to 2 of the integral from two X squared to eight of f of X. Y. Dy Dx. So we can read off from our interior integral that we have Y is bounded below by two X squared And bounded above by eight. Then from the exterior integral, we can read off that X is going to be bounded below by zero and bounded above by two. So if we if we plot out our region, you can see that this is the result. The blue curve down below is y equals two X squared. The orange curve up above is y equals eight. So our integral will be the interior of this region.

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