For converting

integral from cylindrical coordinate to rectangular coordinate first i find the limits(upper/lower boundaries of x,y and z).

We know that (from the integral)

so we can say the upper limit of is:

and lower limit of is:

Also we know from the integral that

So we can say:

And:

As a result the upper limit of is:

Again from the integral

Using the same procedure we can say lower limit of is:

Upper limit of is

and lower limit of is

Now my result is:

But the answer is different. It is:

Why the answer is different? Can anyone kindly point out where i did wrong?