Sect 6 2 Volume Washer Method What if

Sect. 6. 2 Volume (Washer Method)

What if the slices are not solid? Consider the area between two functions rotated about the axis. f(x) g(x) a How can we find the volume? b

Volumes By Washers Similar to when we found the area between curves, we will find the volume of the outer function and inner function using the disk method and then subtract the result. Outer Function Inner Function

(Horizontal Axis of Revolution) Volume = r

Example 1 Find the volume of the solid formed by revolving the region bounded by the graphs of about the x-axis.

You Try… Find the volume of the solid generated when the region between the graphs of the equation and over the interval [0, 2] is revolved about the x-axis.

Perpendicular to the y-axis When revolving the region about the y-axis, we use a similar formula.

Example 2 Determine the volume of the solid obtained by rotating the portion of the region bounded by and that lies in the first quadrant about the y-axis.

Example 2 cont.

You Try… Determine the volume of the solid generated by the region between y = x 2 and y = 4 x, revolved about the y-axis Radius of inner circle? f(y) = y/4 Radius of outer circle? Limits? 0 < y < 16

Other Axes of Revolution Determine the volume when the region is spun about ANY horizontal or vertical axis. Spin about y = c: Remember, you must look at the outer 2 – inner 2 with respect to the axis of revolution. 1. Above the region 2. Between 3. Opposite side of axis y=c a b NOTE: The axis of revolution cannot pass through the region you plan to revolve.

Example 3: Find the volume obtained by rotating the graphs of f(x) = 9 – x 2 and y = 12 for 0 ≤ x ≤ 3 about the line a) y = 12 and b) y = 15.

Example 4: Find the volume of the “wedding band” obtained by rotating the region between the graphs of f(x) = x 2 + 2 and g(x) = 4 – x 2 about the horizontal line y = -3.

You Try… Determine the volume of the solid obtained by rotating the region bounded by y = x and , about y = 1.

Example 5: Find the volume of the solid obtained by rotating the region under the graph of f(x) = 9 – x 2 for 0 ≤ x ≤ 3 about the vertical axis x = -2.

You Try… The region bounded by and about the line x = 2. Find the volume. r R is revolved

More Practice

1. The region bounded by the parabola y = x 2 and the line y = 2 x in the first quadrant is revolved about the y -axis to generate a solid. Find the volume of the solid. y = 0, y = 4

2. The region bounded by the curve y = x 2 +1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid of revolution. x 2 + 1= - x + 3 x 2 + x – 2=0 ( x+ 2 )(x – 1) = 0 x = -2 x=1

3. Find the volume when you spin the region enclosed by y = x 2 , y = 0 and x = 2 about each horizontal line. (Set up the integral, then use your calculator. ) a) y = 8 b) y = 4 c) y = -2

4. Find the volume when you spin the region enclosed by y = x 2 , y = 0 and x = 2 about each vertical line. (Set up the integral, then use your calculator. ) a) x = 4 b) x = 2 c) x = -2

5. Find the volume when the region bounded by y = 6 – 2 x – x 2 and y = x + 6 is revolved about a) x-axis b) y = 3 c) y = 7

Challenge however, r = ? washer disk 2 integrals are needed

disk washer

Closure Explain the difference in finding the volume of a surface of revolution using the Disk and Washer Method. Give the formula for each and explain the variables.