Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y=x^2 , y=0 , x=1 , x=2

Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y=x^2 , y=0 , x=1 , x=2

amistre64 · Answer

if we are doing shells we will need to know the formula for it

agent_a · Answer

This is the equation that I used: $$\int\limits_{1}^{2}2\pi(2-x)(x^2) dx$$ ...based on the formula: $$\int\limits_{a}^{b}2\pi xf(x)dx $$

agent_a · Answer

I solved the problem already. I just wanted to know if my answer is correct or not.

agent_a · Answer

Cylindrical shells have a standard formula, depending on the axis of rotation. If the rotation is done about the y-axis or x=*something* , the radius is in terms of x (vertical cylinder):

$$\int\limits_{a}^{b}2\pi xf(x)dx$$

If the rotation is done about the x-axis or y=*something*, the radius is in terms of y (horizontal cylinder):

$$\int\limits_{c}^{d}2\pi yg(y)dy$$

zepdrix · Answer

Hmm the radius of your shell looks strange.. $\large (2-x)$.
Wouldn't it simply be $\large x$ since we're spinning from the y-axis?

amistre64 · Answer

i tend to go about this as:$$2\pi ~rh$$is the area of one of the shells and then define h and r in their proper terms

amistre64 · Answer

y=x^2 , y=0 , x=1 , x=2

|dw:1376508664437:dw|