Volumes+of+Solids+of+Rotation

 Volumes of Solids of Rotaton Revolving a plane figure or function about an axis or line generates a solid figure with a volume. In the picture above, a function has be rotated around the x-axis, generating a donut-shaped solid.

To find the Volume of a solid generated by rotation, the following equation may be used, by plane splicing: Where...radius or f(x) is the equation of the curve; ...//a// and //b// are the limits of the area being rotated; ...dx shows that the area is being rotated about the x-axis

**Example 1:** Equation:  When given the equation, it is helpful to graph it and sketch the region and solid obtained: Remember to always slice perpendicular to the axis of rotation! It is helpful to draw the disk or washer in addition to sketching the solid:



Now we can see that our width is dx (since the axis of rotation is the x-axis) and the height is the y or f(x) (since the y is perpendicular to the axis of rotation).

After this step, we would normally determine the limits of integration, or where the slices start and stop, but it has been give to us: 1 and 4 in this example.

From here, we can add up all the slices to determine the volume using our equation: Another helpful equation is:  This is used when you are trying to find the volume of the an area enclosed by two curves (y1 and y2) that has been rotated about an axis. It essentially takes the solid formed by the first curve and subtracts the solid formed by the second.

 Equation: //y// = 2//x//2 ; //y// = //x// + 1 ......where x ≥ 0 (assume measurements are in cm)
 * Example 2:**

From the given information we know that the lower limit of integration is x = 0. To get the upper limit of integration we can set the equations equal to each other and solve for x: 2//x^//2 = //x// + 1  2//x^//2 − //x// − 1 = 0 (2//x// + 1)(//x// − 1) = 0 x = 1; -1/2 ....Since we're only looking at x ≥ 0, we can ignore x = -1/2, so the upper limit of integration is x = 1;

We can continue solving with the new equation we learned to find the Volume:

Don't forget to put units! It is helpful to do even if you do not know the units...you can just write u^3 or units^3.

Mr. Clark's Class notes from 11/23/2009 [] []
 * Sources:**

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