Changing the Order of Integration for Double Integrals

[yzc717]

Homework Equations

The Attempt at a Solution

that is $$\int_0^{\sqrt{\pi}}\int_x^{\sqrt{\pi}} \sin(y^2) ~dy ~dx$$

Reverse the order of the integrals (which is possible since the integrand is positive) :
$$0\leq x\leq y\leq \sqrt{\pi} \Rightarrow$$ y ranges from 0 to $$\sqrt{\pi}$$

$$0\leq x\leq y\leq \sqrt{\pi} \Rightarrow$$ x varies from 0 to y.

So the integral is now :

$$\int_0^{\sqrt{\pi}}\left(\int_0^y \sin(y^2) ~dx\right)~dy$$

$$=\int_0^{\sqrt{\pi}}\left(\sin(y^2)\int_0^y dx\right)~dy$$

 

[Random Variable]

Because you changed the order of integration, you don't have to compute the integral of sin(y^2) but rather y*sin(y^2), which can be done by a simple substitution.