Double integral from y to 1 and 0 to 1

[Locoism]

Homework Statement

Evaluate the integral $\int$0 to 1$\int$y to 1$\frac{1}{1+x^4}$dxdy

The Attempt at a Solution

I managed to do the first one, from y to 1, using partial fractions and then some substitution, and I get a huge answer involving some logarithms and arctans that don't simplify. I've tried putting it into some math software to get an idea of what my answer could look like, but the results are hugh ugly formulas that a human being just couldn't do (or at least not this one). Obviously I am doing something wrong. Is there a way to evaluate without computing the indefinite integrals? Or maybe to switch the x and y somehow? Could it be put into polar form? I really don't know what to do here, I've spent hours computing the first integral only to end up with something harder...

 

[Dick]

Interchange the order of integration. Draw a sketch of the region of integration and figure out the x and y limits you need to do that.

 

[iceblits]

swithing the limits results in the same integral and your answer will be that big ugly thing..i think

nevermind..listen to what Dick said

 

[Locoism]

How do I do that? I sketch 1/(1+x^4) and find what?

 

[Dick]

How do I do that? I sketch 1/(1+x^4) and find what?

No, not the function. Sketch the region of integration in the x-y plane. For every value of y, x goes from y to 1. What's the shape of the domain of integration? It's a triangle, isn't it? You want to express the integral dydx instead of dxdy.

 

[Locoism]

oh so the area I'm sketching is the triangle between the line x=y and the x-axis (up to 1). But I don't understand what this integral symbolizes. Am I finding the area in the intersection of that triangle and the function above?

 

[Dick]

oh so the area I'm sketching is the triangle between the line x=y and the x-axis (up to 1). But I don't understand what this integral symbolizes. Am I finding the area in the intersection of that triangle and the function above?

That triangle is the region of integration. You know how to go from that triangle to the given double integral expression, right? Now write down another double integral where you integrate over dy first then dx. The opposite order to the given integral. This is called 'changing the order of integration'. You'll find it's a LOT easier to integrate.

 

[Locoism]

Ok thanks I'll figure it out.
*sigh* I love it when assignments include material that hasn't been mentioned in class...

 

[SammyS]

Ok thanks I'll figure it out.
*sigh* I love it when assignments include material that hasn't been mentioned in class...

It gives you a chance to learn for yourself. That can be a valuable skill to acquire!