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## Homework Statement

This is an exercise from Taylor & Mann page 468 Q5 :

Use the transformation x=au and y=bv to map the region R defined by [itex]\frac{x^2}{a^2} + \frac{y^2}{b^2} ≤ 1[/itex] onto the uv plane.

Evaluate : [itex]\int \int_R \frac{x^2}{a^2} + \frac{y^2}{b^2} dxdy[/itex]

with the aid of this transofrm and polar coordinates.

## Homework Equations

[itex]\int \int_R F(x,y) dxdy = \int \int_{R'} G(u,v)|J| dudv[/itex]

Where |J| is the Jacobian.

## The Attempt at a Solution

So if R is defined to be [itex]\frac{x^2}{a^2} + \frac{y^2}{b^2} ≤ 1[/itex], then using the transformation x=au and y=bv we define a new region R' by [itex]u^2 + v^2 ≤ 1[/itex]

Now I can easily set up a Cartesian integral in terms of u and v, but the point is to use polars to simplify things.

So let u = rcosθ and v = rsinθ and hence R' becomes [itex]r ≤ 1[/itex] since r>0 for [itex]0 ≤ θ ≤ 2π[/itex]

The Jacobian of polars is just r, so J = r.

Using all this information, our integral becomes :

[itex]\int_{0}^{1} \int_{0}^{2π} r^3 dθdr = \pi/2[/itex]

I'm getting the feeling I'm missing something here as the answer at the back of the book is πab/2 which sadly my integral almost evaluates to, but not quite.

Any pointers?