Computer the Volume of a region bounded by 3 curves

[RJLiberator]

Homework Statement

Let R be the region in the first quadrant bounded by all three of the curves x = 2, y = 1, and y = (x−4)^2.
Compute the volumes V1, V2, and V3 of the solids of revolution obtained by revolving R about the x-axis, the y-axis, and the x = 5 line, respectively.


FIRST, I am trying to conceptualize this problem. I have the 3 necessary curves graphed. A 'triangle' looking figure is formed between y=1 to 4 and x =2 to 3. Do I need to find the volume of THIS figure revolved around the various axis points OR do I need to find the volume between y=0 to 1 and x = 2 to 4.

Both areas are bounded by all 3 curves. My intuition tells me to take the volume of the 'triangle' looking figure, but I did not want to proceed until I figured this part out.

 

[RJLiberator]

**FIRST POST AND THIS POST ARE TWO SEPARATE PROBLEMS**

The initial problem (previous problem) states:

Let R be the region in the first quadrant bounded by y = 1 − x2. Compute
the volume V of the solid of revolution generated by revolving R about the x-axis by using
(a) slices
and
(b) shells.
Please verify that you obtain the same value of the volume V by method (a) as by method (b).
Use the calculation above to find the volume of the ball of radius 1 (in 3-dimensional space).

Referring to the bold statements, wouldn't the volume of the radius 1 ball be equivalent to the volume of the region (what I had calculated for that problem) since the bounds are from 0 to 1?

 

[LCKurtz]

Homework Statement

Let R be the region in the first quadrant bounded by all three of the curves x = 2, y = 1, and y = (x−4)^2.
Compute the volumes V1, V2, and V3 of the solids of revolution obtained by revolving R about the x-axis, the y-axis, and the x = 5 line, respectively.


FIRST, I am trying to conceptualize this problem. I have the 3 necessary curves graphed. A 'triangle' looking figure is formed between y=1 to 4 and x =2 to 3. Do I need to find the volume of THIS figure revolved around the various axis points

Yes.

OR do I need to find the volume between y=0 to 1 and x = 2 to 4.

Both areas are bounded by all 3 curves. My intuition tells me to take the volume of the 'triangle' looking figure, but I did not want to proceed until I figured this part out.

Your first thought is correct. $y=0$ is not given as a boundary so your second interpretation is wrong.

 

[LCKurtz]

**FIRST POST AND THIS POST ARE TWO SEPARATE PROBLEMS**

You should start a new thread with a new problem.

 

[STEMucator]

Here's a visual to help you conceptualize:

The only region that is bounded by all three curves is the one depicted above.

If you rotate this region about the x-axis, what is the volume element $dV$ you would choose to integrate?

I believe choosing vertical slices will produce two integrals. Horizontal slices appear to be better as you will require only one integral.

 

[LCKurtz]

Here's a visual to help you conceptualize:

View attachment 73143

The only region that is bounded by all three curves is the one depicted above.

If you rotate this region about the x-axis, what is the volume element $dV$ you would choose to integrate?

I believe choosing vertical slices will produce two integrals. Horizontal slices appear to be better as you will require only one integral.

That is not the correct area. Also you should quote at least part of the post to which you are replying.