Sketch Region of Integration, Reverse Order, Confirm Equality

[kieranl]

Homework Statement

sketch the region of integration and write an equivalent integral with order of integration reversed. Then evaluate both integrals to confirm their equality

Homework Equations

\int\intdydx for 0<=x<=1 and 0<=y<=\sqrt{x}

The Attempt at a Solution

i rearanged the limits so the equation becomes
\int\intdxdy for 0<=x<=y² and 0<=y<=1

but my calculations for the first equation came to 2/3 and the second equation came to 1/3

plus the answer is 1/6 so I am obviously doing something wrong. can someone help? have i even written the second equation right?

 

[arunbg]

Did you first sketch the region of integration? The limits for the second integral(with order of integration reversed) should be taken such that you still integrate over the same region. Do you know how to do this?

 

[kieranl]

yeh I am pretty sure i drew it properly??

 

[arunbg]

Since your attachment is pending approval, assuming your diagram is correct, think about the new set of limits you have to use when you change the order of integration. How do x and y vary now?