Written Homework 2 Hints

1.  First off, if you look at this problem the right way it turns becomes very similar to one we worked out in class and was on the second test. Here is a similar problem.

$$\int_0^1e^{\mbox{max}\{x,1-x\}}dx$$.

First we'll split up the region of integration into two pieces, one where $x$ is larger and one where $1-x$ is larger. Observe that $1-x$ is larger on the interval $[0,1/2]$ and $x$ is larger on the interval $[1/2,1]$. So we can write

$$\int_0^1e^{\mbox{max}\{x,1-x\}}dx=\int_{0}^{1/2}e^{1-x}dx+\int_{1/2}^1 e^xdx,$$

which are fairly simple.

2.  The hint written on the assignment is sufficient.

3.  Use the formula for 4 dimensional volume(similar to the one for area).

$$V(D)=\iiiint_{D}dV,$$ 

and use change of variables to write 

$$dV=\left|\frac{\partial(x,y,z,w)}{\partial(r,\theta,s,\varphi)}\right|,$$

and use the technique here to take the determinant. Be careful about the bounds of integration for $r$ and $s$.

Thats all for now, we'll have the technology to solve 4 and 5 after this week.