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Determine the region $R$ determined by the given double integral.$$\int_{0}^{36} \int_{\frac{y}{6}}^{\sqrt{y}} f(x, y) d x d y$$

Same as 21

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

Johns Hopkins University

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

12:15

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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for this problem, we are asked to determine the region are determined by the double integral from in the interior. We have X from Y over six up to the square root of Y. And then in the exterior we have Y from 0 to 36. So we can sort of naively just say, well X clearly is bounded below by Y over six And bounded above by the square root of why and why is bounded below by zero. And it's bounded above by 36. But if we want to understand what's going on here a little bit better, we're kind of used to having wise a function of X. You can get that The lower boundary would be, well, we have When we're at the lower boundary we have Y over six equals x. Which then means that Y equals six X. At the lower boundary. Note that this is now the lower boundary for why? And at the upper or um that actually will, we'll get to that in a second. Yeah. So we have Y equals six X. And then at the other boundary we'd have X equals root. Why? Which means that why equals X squared. So we can see that. Um And we need to be careful here Using these definitions, if we have that wise between zero and 36, we would then have that correspondingly X must be between zero and 6. We have zero is less than or equal to X is less than or equal to six. And why is going to be greater than or equal to X squared and less than or equal to six X. So we can actually tell then that we have the same region as what we saw in the previous problem.

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