7.4 Day 2 Surface Area

Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

Surface Area:

ds

r

Consider a curve rotated about the x-axis:

The surface area of this band is: 2 r ds

The radius is the y-value of the function, so the whole area is given by: 2

b

ay ds

This is the same ds that we had in the “length of curve” formula, so the formula becomes:

Surface Area about x-axis (Cartesian):

2

2 1 b

a

dyS y dx

dx

To rotate about the y-axis, just

reverse x and y in the formula!

44

3y x Example:

Rotate about the y-axis.

44

3y x

33

4y x

33

4x y

3

4

dx

dy

Example:

Rotate about the y-axis.

3

4

dx

dy

24

0

3 32 3 1

4 4S y dy

4

0

3 252 3

4 16y dy

4

0

3 52 3

4 4y dy

4

2

0

5 33

2 8y y

56 12

2

56

2 15

33

4x y

44

3y x Example:

Rotate about the y-axis.

3

4

dx

dy

15

s . .S A rs

3 5

15

From geometry:

Example: y x rotated about x-axis.

29

02 1

dyS y dx

dx

117.319

29

0

12 1

2S x dx

x

9

0

12 1

4S x dx

x

The TI-89 gets:

Example: The Area of a Surface of Revolution

Find the area of the surface formed by revolving the graph

of f(x) = x3 on the interval [0, 1] about the x-axis, as shown

in Figure 7.46.

Figure 7.46

Solution:

The distance between the x-axis and the graph of f is r(x) = f(x), and because f'(x) = 3x2, the surface area is

Example: The Area of a Surface of Revolution

Find the area of the surface formed by revolving the graph

of f(x) = x2 on the interval [0, ] about the y-axis, as shown. 2

Solution: The Area of a Surface of Revolution

In this case, the distance between the graph of f and the y-axis is

Using the surface area is:

13.614.s

.r x x 2 ,f x x

2 2

02 1 2s x x dx

13

3s

Homework:7.2 day 2: MMM pgs. 59 & 62.