What is the Best Approach for Solving a Complicated Double Integral?

[Hernaner28]

Homework Statement

I've got to calculate:

\displaystyle\int_0^1\displaystyle\int_0^x \sqrt{4x^2-y^2} dy dx

Homework Equations

The Attempt at a Solution

I've tried the change of variable:
\displaystyle t=4{{x}^{2}}-{{y}^{2}} but it doesn't get better. I've also tried polar coordinates but it is not convinient either. Do you know a convenient change? I've been trying to figure it out for a long time.

** I also tried changing the order of integration but no results.


Thanks!

 

[Ray Vickson]

Homework Statement

I've got to calculate:

\displaystyle\int_0^1\displaystyle\int_0^x \sqrt{4x^2-y^2} dy dx

Homework Equations

The Attempt at a Solution

I've tried the change of variable:
\displaystyle t=4{{x}^{2}}-{{y}^{2}} but it doesn't get better. I've also tried polar coordinates but it is not convinient either. Do you know a convenient change? I've been trying to figure it out for a long time.

** I also tried changing the order of integration but no results.Thanks!

The y-integral (for any fixed value of x) is of the form
\int \sqrt{a^2 - y^2} \, dy,
where it happens that a = 2x. This is a standard integral that you must surely have seen before; if not, do a change of variables using a trigonometric substitution.

RGV

 

[Hernaner28]

Thanks! I'll try!