# Use the given transformation to evaluate the integral. \int_R \int 4xy dA where R is the region...

## Question:

Use the given transformation to evaluate the integral.

{eq}\displaystyle \int_R \int 4xy \: dA {/eq} where {eq}\displaystyle R {/eq} is the region in the first quadrant bounded by the lines {eq}\displaystyle y = \frac{2}{3} x {/eq} and {eq}\displaystyle y = 2x {/eq} and the hyperbolas {eq}\displaystyle xy = \frac{2}{3} {/eq} and {eq}\displaystyle xy = 2; \: x = \frac{u}{v}, \: y = v. {/eq}

## Using the Jacobian to Evaluate a Double Integral:

The Jacobian of {eq}x {/eq} and {eq}y {/eq} with respect to {eq}u {/eq} and {eq}v {/eq} is given by:

{eq}J =\dfrac{\partial (x,y)}{\partial (u,v)}=\begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix}\\ {/eq}

We can use the Jacobian to change variables in a double integral by using the formula:

{eq}\iint_{R}f(x,y)dxdy=\iint_{D}f(g(u,v), h(u,v))\left | J \right |dudv {/eq}

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We have to solve the integral {eq}\int \int_R 4xy dA, {/eq}

where R is the region in the first quadrant bounded by the lines {eq}y =... Evaluating Definite Integrals Using the Fundamental Theorem

from

Chapter 16 / Lesson 2
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The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.