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

Bounded by $y=x^{2}$, and $y=\sqrt{x}$, see Fig. Ex. 14

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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For this problem, we are asked to determine the region are determined by the double integral over the region, or the double integral 0 to 1 of the integral from X squared to root X. Of F X, Y D Y D X. So we can then read off that the interior in general over Y from X squared up to root X. So we have that X squared is less than or equal to Y. It's less than or equal to root X. And then from our exterior integral we have that X is going to be between zero and one. So the region determined, then is this here interior where we have? The upper bound is Y equals root X. Lower is Y equals X squared, and we have X is between zero and X equals one.

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