Volume+by+Shells+(BC)

Introduction for Volume by Shells: **Verbally:**
==== .and the derivative of volume of a shell is ==== >
 * ====Is used to calculate the volume of a solid of revolution.====
 * ====Uses cylinder shells====
 * ====When using volume of shells you have the shape of a cylinder but when you slice the cylinder you use slices of rectangles.====
 * ====The integration looks like====
 * Question: If the graph of f is bounded by a and b, how would you find the volume of the cylinder shells? You would use the graph, and the equation [[image:http://latex.codecogs.com/gif.latex?2%5Cpi%20%5Cint_%7Ba%7D%5E%7Bb%7Drf(x)dx]] or [[image:http://latex.codecogs.com/gif.latex?2%5Cpi%20%5Cint_%7Ba%7D%5E%7Bb%7Dx(f(x)-g(x))dx]].

**Numerically:** Some questions you will have to answer by using a table, to do this you simply find the volume of each coordinate and then add them all together (technique of the Left Riemann sum).

 * X || 5 || 8 || 11 || 14 || 17 || 20 ||
 * f(X) || 2 || 4 || 6 || 8 || 6 || 4 ||

Given the table, find the value for 6-sub-units.

5 x 2 x 3 = 30 8 x 4 x 3 = 96 11 x 6 x 3 = 198 14 x 8 x 3 = 336 17 x 6 x 3 = 306

30+96+198+336+306=996, the volume would be the volume added up by each rectangular shell.

width = f(x), and height is dx. **V**layer = If you were given the equation, how would you solve it? You would do then solve.
 * Analytically:** You have a cylinder object and you are trying to solve for the volume. When doing this think of it as peeling a layer of an onion. **V**layer = Length x Width x Height. Length is[[image:http://latex.codecogs.com/gif.latex?2%5Cpi%20(r)]]( which is the circumference),


 * Graphically:** The graph given has the function [[image:http://latex.codecogs.com/gif.latex?y=-x%5E2+9x]], rotated around the y-axis, find the volume.



Solution: **V**object=, then solve. (x=radius)

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