Solids of revolution, spring constants, and limits

These are some problems that I'm having trouble with from a review packet. I know the answers, but I don't know how to do them. I would greatly appreciate any help I could get, even if you only help on one or two. I included my work for some of them so you can tell me what I'm doing wrong.

1.) Find lim t --> 0 of [ln(x-9)^8] / [2x - 20]

I used L'hospital's rule a few times and got 0, but the answer is 4. I see 8/2 in the last part, but I thought the ln in the numerator approaches 0...?

http://i42.tinypic.com/205ussp.jpg

2.) The region y = 2x - x^2 and y = 0 is rotated around the x-axis. Find the value of a if the resulting volume is 44π / 3.

The answer is 3, but I get a = 6.5.

http://i42.tinypic.com/10yo39c.jpg

3.) It takes 1/200 Joules to stretch a spring from its natural lenth to another y centimeters. It takes 9/200 Joules to stretch the same spring from its natural length to another y + 4 centimeters. Find the value of y. What is the value of the spring constant?

y = 2 and the spring constant is 25. I don't know how to start.

4.) The region bounded by y = 0, x = 0, and y= (3-x) / π is rotated around the y-axis to form a solid of revolution. use the disc method to find the volume.

The answer is 9. I was able to do it with the shell method, but when I used the

disc method, I got some ridiculous answer that included π ^2.

5.) Suppose it is always true that 2 ≤ f' (x) ≤ 4. What is the smallest possible value of f (7) - f (2)?

The answer is 10.

Re: Solids of revolution, spring constants, and limits

That's because you interpret the limit incorrect (and I admit it's not stated very clear!), if the answer is 4 then the limit should be:

Do you note the difference?

I wonder why you let in stead of , probably a typo ... ?

Re: Solids of revolution, spring constants, and limits

Re: Solids of revolution, spring constants, and limits

Quote:

Originally Posted by

**trevor22** These are some problems that I'm having trouble with from a review packet. I know the answers, but I don't know how to do them. I would greatly appreciate any help I could get, even if you only help on one or two. I included my work for some of them so you can tell me what I'm doing wrong.

1.) Find lim t --> 0 of [ln(x-9)^8] / [2x - 20]

I used L'hospital's rule a few times and got 0, but the answer is 4. I see 8/2 in the last part, but I thought the ln in the numerator approaches 0...?

http://i42.tinypic.com/205ussp.jpg
2.) The region y = 2x - x^2 and y = 0 is rotated around the x-axis. Find the value of a if the resulting volume is 44π / 3.

The answer is 3, but I get a = 6.5.

http://i42.tinypic.com/10yo39c.jpg
3.) It takes 1/200 Joules to stretch a spring from its natural lenth to another y centimeters. It takes 9/200 Joules to stretch the same spring from its natural length to another y + 4 centimeters. Find the value of y. What is the value of the spring constant?

y = 2 and the spring constant is 25. I don't know how to start.

4.) The region bounded by y = 0, x = 0, and y= (3-x) / π is rotated around the y-axis to form a solid of revolution. use the disc method to find the volume.

The answer is 9. I was able to do it with the shell method, but when I used the

disc method, I got some ridiculous answer that included π ^2.

5.) Suppose it is always true that 2 ≤ f' (x) ≤ 4. What is the smallest possible value of f (7) - f (2)?

The answer is 10.

For 2. what is "a" supposed to represent?

For 4. note that the region you are rotating is a triangle, which means that the region will be a right-angle cone 3 units in radius, units in height.

So the volume will be

Anyway, to use the discs method, visualise the area of that triangle being approximated using horizontal rectangular strips. They will have a length = x, and a width (some small change in y.

When you rotate these strips, you get cylinders (discs), that will have radius = x and height

So the volume of each disc is .

So the volume of your region can be approximated by summing these cylinders.

As you increase the number of strips and make , the sum converges on an integral, and the approximation becomes exact. Note that your bounds will be y bounds...

as required.

Re: Solids of revolution, spring constants, and limits

Thanks. I was able to get all of them now except for 3. I wrote it wrong up there. It's supposed to say.

2. The region between y = 2x - x^2 and y = 0 is rotated around the line x =a. Find the value of a if the resulting volume is 44π / 3.

The answer is 3, but I get a = 6.5.

http://i42.tinypic.com/10yo39c.jpg