Solids of revolution, spring constants, and limits

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December 14th 2011, 09:01 AM
trevor22

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.
December 14th 2011, 09:51 AM
Siron

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:
$\lim_{x\to 10} \frac{\ln[(x-9)^8]}{2x-20}$
Do you note the difference?

I wonder why you let $t\to 10$ in stead of $x\to 10$ , probably a typo ... ?
December 14th 2011, 02:21 PM
skeeter

Re: Solids of revolution, spring constants, and limits

Quote:

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.

(3) $F = kx$

$\int_0^y kx \, dx = \frac{1}{200}$

$\int_0^{y+.04} kx \, dx = \frac{9}{200}$

evaluate both definite integrals using the FTC ... this will give you two equations in k and y which you should be able to solve. Also, y = 2 cm , k = 25 N/m.

--------------------------------------------------------------------------

(4) $y = \frac{3-x}{\pi}$

$x = 3 - \pi y$

$V = \pi \int_0^{3/\pi} (3 - \pi y)^2 \, dy$

--------------------------------------------------------------------------

(5) $2 < \frac{f(7)-f(2)}{7-2} < 4$ ... why?
December 14th 2011, 03:14 PM
Prove It

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, $\displaystyle \begin{align*} \frac{3}{\pi} \end{align*}$ units in height.

So the volume will be

$\displaystyle \begin{align*} V &= \frac{\pi r^2 h}{3} \\ &= \frac{\pi \cdot 3^2 \cdot \frac{3}{\pi}}{3} \\ &= 9 \textrm{ units}^2 \end{align*}$

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 $\displaystyle \begin{align*} = \Delta y \end{align*}$ (some small change in y.

When you rotate these strips, you get cylinders (discs), that will have radius = x and height $\displaystyle \begin{align*} = \Delta y \end{align*}$

So the volume of each disc is $\displaystyle \begin{align*} \pi x^2 \Delta y = \pi \left(3 - \pi y\right)^2 \Delta y \end{align*}$ .

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

$\displaystyle \begin{align*} V &\approx \sum{\pi \left(3 - \pi y\right)^2 \Delta y} \end{align*}$

As you increase the number of strips and make $\displaystyle \begin{align*} \Delta y \to 0 \end{align*}$ , the sum converges on an integral, and the approximation becomes exact. Note that your bounds will be y bounds...

$\displaystyle \begin{align*} V &= \int_0^{\frac{3}{\pi}}{\pi\left(3 - \pi y\right)^2\,dy} \\ &= -\int_3^{0}{u^2\,du} \textrm{ after making the substitution }u = 3 - \pi y \implies du = -\pi\,dy \\ &= \int_0^3{u^2\,du} \\ &= \left[\frac{u^3}{3}\right]_0^3 \\ &= \frac{3^3}{3} - \frac{0^3}{3} \\ &= 9 \end{align*}$

as required.
December 15th 2011, 09:12 AM
trevor22

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