Reflection

In class, we have been learning about solving for volume after graphing a washer or a disk. Volume > S area of cross section.
Disk Method = (pi)S(r^2)dx
Washer Method = (pi)S(top^2)-(bottom^2)
When you revolve a graph, then you spin it around on the x-axis.

EXAMPLE: Find the volume of the solid formed by revolving the region bounded by the
graphs of y = (squareroot of x) and y = x^2 about the x-axis.
First, you plug in the 2 equations into your calculator and graph them.
After you have graphed then you draw the graph reflection to determine if it
is a washer or a disk. For this particular graph, it is a washer.
Set the 2 equations equal to find the bounds.
(squareroot of x)^2 = (x^2)^2
x = x^4
x^4 - x = 0
x(x^3 -1) = 0
x = 0, 1
Plug into formula.
(pi)1/S/0 (squareroot of x)^2 - (x^2)^2
(pi)1/S/0 x - x^4
Find the integral.
pi[1/2x^2 - 1/5x^5]
Plug in the bounds.
(pi)[1/2(1)^2 - 1/5(1)^2] - [1/2(0)^2 - 1/5(0)^2]
= 3pi/10

EXAMPLE: Find the volume of the solid formed by y = 4 - x^2 revoved about the x-axis
in Quadrant I.
First graph the equation.
(pi)2/S/0(4-x^2)dx
(pi)2/S/0(16-2x^2+x^4
Find the integral and plug in bounds.
(pi)[16(2)-2/3(2)^3+1/5(2)^5] - [16(0)-2/3(0)^3+1/5(0)^5]
= 256pi/15