Volumes of a Region bounded by Two Curves

[doublehh06]

1. Let R be the region bounded by the curves y = x² and y = x + 2.
(a) Sketch the region R and label the points of intersection between the two curves.
(b) Suppose we rotate R about the x-axis. Compute the volume of the resulting solid.
(c) What is the volume of the solid obtained by rotating the region R about the line x = 2?

Homework Equations

We need to know the equation for solving volume using a definite integral with the disc method and cylindrical method.

The Attempt at a Solution

I have part (a). For part (b), I set up the problem using the disc method which is the definite integral from [-1,2] of A(x) dx.

The set up looked like The definite integral from [-1,2] of pi(x+2)²dx.
After solving this, I got 21pi but am doubtful of my answer.

For part (c), I set up the problem using the cylindrical method which is the definite integral from [-1,2] of 2*pi*x (x+2)dx with radius x, circumference 2pi x and height x + 2. I got 12pi but am once again doubtful of my answer.

Am I on the right track? Help please?

 

[jackmell]

When you use the "washer" (disc) method, the volume of the washer is:

$$\pi(r_1^2-r_2^2)h$$

where the outer radius, r_1, is the line f(x)=x+2 and the inner radius, r_2, is g(x)=x^2 and the "height" of the infinitesimal disc would be dx since you're integrating with respect to x so the integrand would

$$\pi(f^2-g^2)dx$$

right?

Now for the cylindrical shell method, since you're going around the x-axis, can you see where that would require two integrals? The first "inner" integral would be the shells contributed by just the parabola up to the height where the line f=x+2 intersects it. Once you pass that, then the "height" of the shells would be determined by the distance between the line f=x+2 and the right side of the parabola or:

$$V=I_1+I_2=\int_0^1 (shell1)dy+\int_1^4 (shell2) dy$$

Maybe that's not very clear. How about just look in the calculus textbook on both of these methods. That would help too.

 

[doublehh06]

I have reviewed the topics in the book and feel I have an understanding; however, I have never seen problems like these before and can't relate them to examples in the book. I thought the disc/cylindrical methods may be appropriate, but I am not positive.

When you read the problem itself for the first time, how would you go about solving it?

Thanks.

 

[jackmell]

I have reviewed the topics in the book and feel I have an understanding; however, I have never seen problems like these before and can't relate them to examples in the book. I thought the disc/cylindrical methods may be appropriate, but I am not positive.

When you read the problem itself for the first time, how would you go about solving it?

Thanks.

Hi. I would solve it using the "washer" method. The volume of the infinitesimal washer is:

$$\pi(f^2-g^2)dx$$

right?

So then just integrate over the range -1 to 2.

 

[doublehh06]

That sounds right for part (B); however, would you use the same method for part (c)? Or would you try something else?

 

[jackmell]

Hi Double. Sorry I'm not able to get to the web much these days. Maybe you've solved this already. Anyway, part C would best be solved using cylindrical shells and would only take one integral.