Calc II finding volume of solid by rotating

In summary, to find the volume of a solid obtained by rotating a region bounded by curves about a specified line, use the formula Volume = Integral of A(y) dy, where A(y) = (pi)(r)^2. To determine the radius portion, subtract the distance from the rotating axis (in this case, -1) from the function. This is always done as xright - xleft, so for whichever radius you are using, it will be written as (xright + 1)^2. While it is also possible to subtract the other way, it is recommended to always write it as xright + 1 to develop good habits.
  • #1
MillerGenuine
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Homework Statement



Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=x^2
x=y^2
about x= -1


Homework Equations



Volume= Integral of A(y) dy where A(y)= (pi)(r)^2



The Attempt at a Solution



My question is how to find the radius portion in the (pi)(r)^2
I know that you subtract the inner radius from the outter radius..and the book says that this is
[(y^1/2) - (-1)]^2 - [y^2 - (-1)]^2

I don't understand how they determined that you subtract (-1) from the function, I realize this is the distance from the roatating axis, by why not [(-1) - (y^1/2)]^2 - ...

how do i determine whether i subtract or add the (1) ?
 
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  • #2
The distance horizontally is always xright - xleft. In this case xleft = -1, so for whichever radius you are doing you are going to use

π(xright - (-1))2 = π(xright + 1)2.

Since the radius is squared it would be OK to subtract in the other way, but it is a good habit to always write it this way.
 

1. What is the concept behind finding volume of solid by rotating in Calc II?

In Calc II, finding the volume of a solid by rotating involves using the method of cylindrical shells or the method of disks/washers. This concept relies on revolving a 2-dimensional shape around a specific axis to create a 3-dimensional solid, and then using integration to calculate the volume of that solid.

2. How do I determine the axis of rotation for a given 2-dimensional shape?

The axis of rotation can be determined by visualizing the shape and deciding which direction it will be rotated. For example, if the shape is being rotated horizontally, the axis of rotation will be vertical. In some cases, it may be helpful to sketch the shape and the axis of rotation to better understand the problem.

3. What is the difference between using the method of cylindrical shells and the method of disks/washers?

The method of cylindrical shells involves using vertical strips to create cylindrical shells, while the method of disks/washers uses horizontal slices to create disks or washers. Depending on the shape and the axis of rotation, one method may be easier to use than the other. It is important to understand both methods and choose the one that is most appropriate for the given problem.

4. How do I set up the integral to find the volume of a solid using the method of cylindrical shells?

To set up the integral, you will need to determine the height and radius of each cylindrical shell. The height will be the distance between the top and bottom of the shape when it is rotated, and the radius will be the distance from the axis of rotation to the edge of the shape. The integral will then be the sum of the volumes of each cylindrical shell, with the height and radius values plugged in.

5. What are some strategies for solving tricky problems involving finding volume of solid by rotating?

Some helpful strategies for solving difficult problems include breaking the shape into smaller, more manageable pieces, using symmetry to simplify the problem, and reviewing basic geometry formulas to find the necessary measurements. It can also be helpful to practice with a variety of problems to become more familiar with the concepts and techniques involved.

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