Question

Sketch the region of integration, reverse the order of integration, and evaluate the integral. see attachment.

Answer 1

My main concern is getting the new integral values. I already solved to obtain y^3=x. I've noticed that the graph is going from 0<y<2 after I graphed the whole thing. Also would the new integral values be from 0<x< to y^3?

Answer 2

assuming that 0<y<2 and 0<x<y^3 are indeed the new values I can just integrate with respect to x and then y. But how would I know for sure if I got the right values?

Answer 3

to reverse the limits, i would suggest looking at the graph by rotating your head 90 degrees clockwise, or rotate your paper 90 degrees counterclockwise.

as you can see 0 < y < 2 is correct

however, for the limits of x, when we have a line as a boundary, we use that constant value, when we don't, we use the function either above or below. [my explanation here is really bad lol, but it comes with practice]

so the limits of x are:

y^3 < x <8

Answer 4

I wish the y's would cancel out nicely...

Answer 5

nevermind gtot it. I accidentally combined the 8 and the fraction. It's 1/4ln17

Answer 6

oops I meant 16 - 1/ 4 ln 17