Question

Consider the following.y = 5xe?x, y = 0, x = 2; about the y-axis(a) Set up an integral for the volume V of the solid obtained by rotating the region bounded by the given curve about the specified axis.(b) Use your calculator to evaluate the integral correct to five decimal places

Answer 1

Answer:

The volume for the solid of revolution is $V=\displaystyle 10\pi\int_0^2 x^2e^x\, dx$ and its value is 401.43623.

Step-by-step explanation:

Since the function is rotating over the y-axis and it is a function in terms of x we can use Shell integration which general formula is given by

$V=2\pi \int\limits^a_b {x f(x)} \, dx$

a) Setting up the integral.

We are given one end point for x which is x = 2, but we are not given the other, so we can set y = 0 on the given equation to get:

$0 = 5xe^x$

Notice that the exponential function is not going to be equal to 0, so we get the other endpoint for the interval as

$0 = 5x$

which give us

$x=0$

Thus we integate from x = 0 to 2, so we will get:

$V=\displaystyle 2\pi\int_0^2 x (5x)e^x\, dx$

And we can simplify that to

$V=\displaystyle 10\pi\int_0^2 x^2e^x\, dx$

b) Using a calculator to evaluate the integral.

We can use a calculator to write the integral including the interval to get:

$V = 20 \left(e^2-1\right) \pi$

And its value rounded to 5 decimal places is

$V=401.43623$