# A difficult multiple integral question

1. Apr 30, 2009

### bigevil

1. The problem statement, all variables and given/known data

I don't have the answer for this one, so hopefully someone can help...

(a) By reversing the order of integration, evaluate $$\int_0^8 \int_{y^{1/3}}^2 \sqrt{x^4 + 1} dx dy$$

(b) By evaluating an appropriate double integral, find the volume of the wedge lying between the planes z = px and z = qx (p > q >0) and the cylinder $$x^2 + y^2 = 2ax$$ where a> 0.

Find also the area of the curved surface of the wedge.

3. The attempt at a solution

This is a really strange, and in my opinion, incredibly difficult question. The first part is straightforward.

For the second part, I tried to visualise the wedge in the x-z, x-y and y-z dimensions. I settled for a view in the x-z dimension:

and then did a relative volume. The area enclosed between the lines is

$$\int_0^{2a} \int_{qx}^{px} dy dx = 2a^2 (p-q)$$

then the relative volume is given by $$\pi a^3 (p-q)$$

Can somebody help check this please? I don't think it's the right answer, seeing as it has absolutely nothing to do with the (a) integral, which I'm guessing is a hint of some sort.

How do I work out the second part? (Just some hints please.) I am thinking of parameterisation at the moment (since y is dependent on x, z is dependent on x), although not quite sure how to do that.