Answer:
The volume for the solid of revolution is [tex]V=\displaystyle 10\pi\int_0^2 x^2e^x\, dx[/tex] 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
[tex]V=2\pi \int\limits^a_b {x f(x)} \, dx[/tex]
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:
[tex]0 = 5xe^x[/tex]
Notice that the exponential function is not going to be equal to 0, so we get the other endpoint for the interval as
[tex]0 = 5x[/tex]
which give us
[tex]x=0[/tex]
Thus we integate from x = 0 to 2, so we will get:
[tex]V=\displaystyle 2\pi\int_0^2 x (5x)e^x\, dx[/tex]
And we can simplify that to
[tex]V=\displaystyle 10\pi\int_0^2 x^2e^x\, dx[/tex]
b) Using a calculator to evaluate the integral.
We can use a calculator to write the integral including the interval to get:
[tex]V = 20 \left(e^2-1\right) \pi[/tex]
And its value rounded to 5 decimal places is
[tex]V=401.43623[/tex]
