What Went Wrong with Finding the Area of an Ellipse?

[Knissp]

Homework Statement

Show that the area of $$x^2/a^2+y^2/b^2=1$$ is $$\pi ab$$

Homework Equations

Given transformations:
$$x=au$$
$$y=bv$$

The Attempt at a Solution

$$J(u,v) = a*b$$

$$\int\int ((au)/a)^2+((bv)/b)^2 J(u,v) dudv$$

$$\int\int u^2+v^2 J(u,v) dudv$$

$$\int_0^{2\pi}\int_0^1 r^2 J(u,v) r drd\theta$$

$$\int_0^{2\pi}\int_0^1 a b r^3 drd\theta$$

$$\int_0^{2\pi} 1/4 a b d\theta$$

=$$\frac{\pi a b}{2}$$

But that's obviously wrong. Where did I mess up?

 

[arildno]

Right from the start lies your mistake!
You are to integrate:
$$\int_{A}dA=\int_{A}dxdy=\int_{A}abdudv=\int_{0}^{2\pi}\int_{0}^{1}abrdrd\theta=\pi{ab}$$

 

[Knissp]

I can't believe I even had to ask this! Thanks so much!