Calculating the Volume of a Rotated Solid Using Calculus

[anthonybommarito1]

Find the volume of the solid generated by rotating the region of the x-y plane between the line Y=4,the curve Y=3sin(x)+1 on the interval [-pi/2,3pi/2] about the line Y=4Hi I am having trouble setting up this problem my guess for the integral would be from -pi/2 to 3pi/2 of (4-3sinx+1)^2 because it is being rotated around the line y=4. When i plug the answer i get from this into my course homework site it gives me a hint saying that I should i try use a double angle formula which leads me to believe i may have set of my volume equation wrong. Any help would be greatly appreciated!

 

[Simon Bridge]

Welcome to PF;
Have you tried looking at any of the many tutorials online about how to find a solid of rotation about an off-axis line?

http://www.emathhelp.net/notes/calc...lume-of-solid-of-revolution-about-slant-line/
... for example?

If so then please show your working so far.

 

[anthonybommarito1]

Okay well I know because it is being rotated around a line y=4 that is not on the axis it follows that the integral from a to be which in this case would be -pi/2 to 3pi/2 of the outer radius squared minus the inner radius squared times pi. I believe that 3 sin(x)+1 was my outer radius because it is furthest away from the line y=4, my inner radius is zero because my upper bound is the axis I am rotating about so that gives me the integral from -pi/2 to 3pi/2 of (4-3sinx+1)^2 because the line y=4 is above 3sinx+1 i subtracted it. Distributing the square gives me 9sin^2x -30sinx+25 times pi. I am confused if i set up the correct function to be integrated and I worked on many problems like this, the only difference is that their were two functions both off of the axis of rotation. I am just struggle with the idea that it is on the axis of rotation and how i apply a double angle to that.

 

[Svein]

What if you substitute z=y-4?

 

[Simon Bridge]

There are lots of ways of doing the problem - if the hint makes no sense it may just be that you have used a different approach to what the book's author expected.
A quick way to check your understanding of the problem is to sketch the function and shade in the region being integrated. I would look for an equivalent solid - one with the same volume - that is easier to set the integral up for.

 

[Mark44]

A quick way to check your understanding of the problem is to sketch the function and shade in the region being integrated.

In addition to sketching the region that is being revolved, I always draw a sketch of the solid of revolution, including the typical volume element.

 

[Simon Bridge]

If I'm following things - there is an error here

the integral from -pi/2 to 3pi/2 of (4-3sinx+1)^2

... this does not follow from the description. Check your algebra.