Integrating e^(6x^2+y^2) Over a Circular Disk with Maxima and Minima Estimation

[Larrytsai]

Homework Statement

Using the maxima and minima of the function, produce upper and lower estimates of the integral

int(int) e^(6(x^2+y^2))dA x^2+y^2<= 4 where D is a circlular disk

Homework Equations

The Attempt at a Solution

can someone send me off in the right direction? Like how would i use maxmia and mina of the function??

 

[hunt_mat]

Try converting to polar co-ordinates, $$x=r\cos\theta ,y=r\sin\theta$$. This should allow you to compute the integral explicitly.

Mat

 

[HallsofIvy]

In polar coordinates, the integrand is just $e^{6r^2}$ which has maximum value on the boundary of the circle, where $r= \sqrt{x^2+ y^2}= 2$ of $e^{24}$ and minimum at the center, where r= 0, of $e^0= 1$. That allows you to answer the question of "upper and lower estimates" by just multiplying the max and min by the area of the disk.