Set up a definite integral that represents the volume obtained by rotating the region between the curve y^2=x and y^2=2(x-1) about the lines below.

(a) Rotate about the y-axis

(b) Rotate about the line y=3

Question

Set up a definite integral that represents the volume obtained by rotating the region between the curve y^2=x and y^2=2(x-1) about the lines below. 

(a) Rotate about the y-axis 

(b) Rotate about the line y=3

dido625 · Answer

Normally, I am okay with elementary calculus but I always struggle with this part. Any guidance?

nincompoop · Answer

first, try and draw

dido625 · Answer

I already drew out the region. Looks like a washer to me.

nincompoop · Answer

how do you determine between washer and shell method?

dido625 · Answer

Inner and outer radius for Washer and cylinders.

nincompoop · Answer

or even disk

dido625 · Answer

Yea. Disk too.

nincompoop · Answer

so if you already know that it is washer, then you just have to use the formula

dido625 · Answer

I know. I'm just having writing it out xD xD .Hold on.

dido625 · Answer

$$\int\limits_{a}^{b}\pi(x^2-(2(x-1))^2 dx$$

dido625 · Answer

Is that it? I can easily find a and by setting the two equations equal to each other.

nincompoop · Answer

then do it

dido625 · Answer

$$-\sqrt{2},\sqrt{2}$$ ?