1.(16 pts.) find the volume of the solid obtained by revolving the region enclosed by y = xex , y = 0 and x = 1 about the x-axis.

The Volume is 5.018 cubic units

Explanation:

Volume Of A Solid Of Revolution

Let f(x) be a continuous function defined in an interval [a,b], if we take the area enclosed by f(x) between x=a, x=b and revolve it around the x-axis, we get a solid whose volume can be computed as

It's called the disk method. There are other available methods to compute the volume.

We have

And the boundaries defined as x=1, y=0 and revolved around the x-axis. The left endpoint of the integral is easily identified as x=0, because it defines the beginning of the region to revolve. So we need to compute

We need to first determine the antiderivative

Let's integrate by parts using the formula

We pick

Then

Applying by parts:

Now we solve

Making

Applying by parts again:

The last integral is directly computed

Replacing every integral computed above

Simplifying

Now we compute the definite integral as the volume

Finally

The Volume is 5.018 cubic units