Homework Help: Iterated Integrals

1. Apr 8, 2005

stunner5000pt

i need to solve this
$$\int_{0}^{\sqrt{\pi}} ( \int_{x}^{\sqrt{\pi}} sin y^2 dy) dx$$

now i know i have to change the order of this
the integrand is bounded by the triangle from x = 0 to $x= \sqrt{\pi}$ here's where i am stuck
what is the boundary of the y?? is y bounded below by x=0 and above by x =1??

so what would the limits of integration change to?? (for the inside one from 0 to root pi?) and the outside one stays the same??

pelase help!

2. Apr 8, 2005

ehild

Look at the attached figure. You have to integrate for the yellow triangle, according to the boundaries of your original integral. If you change the order of integration, that is you integrate by x first, it would go from x=0 to x=y; and then by y which goes from y=0 to y= sqrt (pi).

ehild

Last edited: Jun 29, 2010
3. Apr 8, 2005

stunner5000pt

how do you know that it is the upper triangle and not the lower triangle??

4. Apr 8, 2005

whozum

Because in the original integral:

$$\int_{0}^{\sqrt{\pi}} ( \int_{x}^{\sqrt{\pi}} sin y^2 dy) dx$$

Look at the limits for the dy integral, y goes from y=x to y=root pi. If it were the white triangle in the image, then y would be going from 0 to x.

5. Apr 8, 2005

marlon

ehild is completely correct

marlon

6. Apr 8, 2005

dextercioby

You may wanna search mathworld for Fresnel' S(x) antiderivative...

Daniel.

Last edited: Apr 8, 2005
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