Question

Find the volume when the shaded region is rotated 360 degrees about the y axis for (x^2- 6x + 9)

Answer 1

Are you supposed to use calculus?

Answer 2

yes

Answer 3

Well, thats a parabola, so you need to imagina what would happen if you took a infinitesimally small vertical slice of it and revolved it. You would end up with a cilinder of this volume: \[dV=2 \pi xf(x)dx\]Where: 2 pi x is the circumference, f(x) the height and dx the width. Then you just solve for V by suming all of the cilinders that are formed which is: \[V=\int\limits_{0}^{a}2 \pi x f(x)dx=2 \pi \int\limits_{0}^{a}x(x ^{2}-6x+9)dx= 2 \pi \int\limits_{0}^{a}(x ^{3}-6x ^{2}+9x)dx\]\[V=2 \pi (\frac{ x ^{4} }{ 4 }-6\frac{ x ^{3} }{ 3 }+9\frac{ x ^{2} }{ 2 })/_{0}^{a}\]To know a you need to know which is the right end of the parabola. You integrate from zero since you only need the right side because each sliver causes a cilinder that when rotated 360° causes the same cilinder as the sliver in the opposite side, so if you integrate from -a to a, then you would be counting two times the same cilinder.

Answer 4

he graph looks like this. Should I now substitute the x values to find the volume?

Answer 5

A medal Thank you so much!

Answer 6

Oh! It starts at 3! So you need to integrate from 3 to a. The problem is that you still have no limit for a.