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Determine the region $R$ determined by the given double integral.$$\int_{0}^{1} \int_{0}^{3 x^{2}} f(x, y) d y d x+\int_{1}^{24-x^{2}} f(x, y) d y d x$$

Same as 29

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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University of Michigan - Ann Arbor

Lectures

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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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for this problem. We are asked to determine the region are determined by the given double integral. So what I'll note is that we have two terms. I'll label them as A and B. And it looks like something miss copied one second. Yeah, this should be, let me correct this, that second term which I'm calling B should be the integral from 1 to 2 of the integral from zero to X squared. Or excuse me to four minus X squared of f dy dx. So from term A we can read off from that interior integral that we have zero is less than or equal to why is less than or equal to three X squared. And we have that zero is less than or equal to X is less than or equal to one from the exterior. Then from term be we have zero is less than or equal to why is less than or equal to four minus X squared. And from the exterior integral we can see that we have one is less than or equal to X is less than or equal to two. So if we plot out the region that we are integrating over here, one moment you make a correction here, we'll have that what our result is. Let's see here. So from 0 to 1 we want to be integrating why from 0 to 3 X squared. So we would have that this area here is region A. From that first term and then the remaining part is bounded above by four minus X squared. So this is region BB

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