Shells Method Issues: between y = 2x, y = x^2, about y = -1

[clmason]

Hello everyone,
I am working on a calculus assignment and unfortunately my professor didn't spend much time going over how to do the Shell and Washer methods. In particular I am having some trouble on two of the problems.

1) Let R be the region between y= 2x and y= x^2. Set up and evaluate the integral necessary to compute the volume when R is rotated about the given line using the given method.
Line = y= -1 Method = Shells

2)Let R be the region between y= 2x and y= x^2. Set up and evaluate the integral necessary to compute the volume when R is rotated about the given line using the given method.
Line = y= -1 Method = Washers

Thanks for the help

 

[galactus]

Re: Shells Method Issues

For #1:

Since we are using shells and revolving about y=-1, we want to have our integral in terms of y.

Because when using shells, the cross sections are parallel to the axis we are revolving about.

See?. Picture the line y=-1 drawn on the graph. Now, if the cross sections are parallel to it they are 'stacked up' the y axis.

The limits of integration can be found by solving the given functions for y and setting equal: \(\displaystyle \frac{y}{2}=\sqrt{y}\). y=0 and 4.

\(\displaystyle 2{\pi}\int_{0}^{4}(1+y)(\sqrt{y}-\frac{y}{2})dy\)

#2: With washers and get the same. The cross-sections are perpendicular to y=-1 and we keep it in terms of x and integrate from 0 to 2.

\(\displaystyle 2x=x^{2}\), x=0 and 2

\(\displaystyle {\pi}\int_{0}^{2}\left[(2x+1)^{2}-(x^{2}+1)^{2}\right]dx\)

 

[clmason]

Another shells issue:

y=e^x, y=e^-x, x=0 and x=1 about the y-axis.

Thanks for any help.

 

[Deleted member 4993]

Another shells issue:

y=e^x, y=e^-x, x=0 and x=1 about the y-axis.

Thanks for any help.

Please start a new thread with a new problem.

For this problem, did you follow Galactus's advice?

1)Did you sketch the functions in the problem?

2)Did you establish the boundaries of the region?

3)Did you sketch the elementary shell in your earlier sketch (step 1)?

4)Did you find the volume of the elemantary shell?

5)Did you find the limits of integration from step 2 above?

6) Did you finish the integration - and find the volume of the rotated region?

Exactly where are you stuck?

 

[clmason]

I'm sorry I will start a new thread next time.

I am stuck in finding the limits of integration,I know it will be 2pi integral from the limits of integration ( ) (outside-inside) dy but I am getting a little mixed up here with the final integration.

Thanks,
-CM