Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y=4x-x^2 y=3 about x=1

Answer 1

Answer:

$\frac{16\pi}{3}$.

Step-by-step explanation:

I graphed the region in the image below. The blue line is y=3, the purple line is x=1 and the green curve is y = $4x-x^{2}$. The shaded region in blue is the region we are going to rotate.

Now, to find the volume  v= $2\pi \int\limits^a_b {p(x)h(x)} \, dx$ where a=1, b=3 (left and right points of the region), p(x) is the distance from the rotation axis to the diferential Δx, we say p(x)=x and h(x) is the height of the region, in this case is h(x)= $4x-x^{2}-3$. Then,

v =  $2\pi \int\limits^1_3 {x(4x-x^{2}-3)} \, dx$

=  $2\pi \int\limits^1_3 {4x^{2}-x^{3}-3x} \, dx$

= $2\pi (\frac{4x^{3}}{3}-\frac{x^{4}}{4}-\frac{3x^{2}}{2})^{3}_1$

= $2\pi (36-\frac{81}{4}-\frac{27}{2}-\frac{4}{3}+\frac{1}{4}+\frac{3}{2})$

= $2\pi*\frac{8}{3}$

=  $\frac{16\pi}{3}$.