8.2 Area of a Surface of Revolution by SGLee - HSKim, JYLee, JHLee
1-2. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.
1. 
, 
; 
-axis
.
Calculus-Sec-8-2-Solution
8.2 Area of a Surface of Revolution by SGLee - HSKim, JYLee, JHLee
1-2. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.
1. , ; -axis
.
2. 
, 
; 
-axis
(range from a to b) Let, 
=> 
=> 
=> 
( or 
)
3-7. Find the area of the surface obtained by rotating the curve about the 
-axis.
3. Find the area of the surface when 
goes to 0.
, 
.
http://matrix.skku.ac.kr/cal-lab/cal-8-2-3.html
4. Find the area of the surface when 
is a positive even integer.
. 
.
http://matrix.skku.ac.kr/cal-lab/cal-8-2-4.html
5. 
, 
6. 
, 
The surface obtained by rotating the curve about the 
-axis is
Substitute 
,
=> 
Let 
and 
, then 
.
Calculate and simplify to have 
.
7. Find the area of the surface obtained by rotating the curve about the 
-axis.
, 
.
http://matrix.skku.ac.kr/cal-lab/cal-8-2-7.html
From the above figure, it is clear that the maximum is circle of unit radius in the first quadrant.
When we rotate the unit circle in the first quadrant about the x-axis we obtain half unit sphere. Hence the required area is 
.
8-10. The given curve is rotated about the 
-axis. Find the area of the resulting surface.
8. 
, 
.
9. 
, 
.
10. 
, 
The surface obtained by rotating the curve about the 
-axis becomes
(Range form a to b)
Then substitute 
to above equation.
(
, then 
(because of 
) and 
so 
)
.
11. Find the area of the surface of the solid of revolution obtained by rotating about the 
-axis the circle
, 
.
The right semicircle and left semicircle equations are given by
and 
, respectively.
.
12. Use a CAS to find the area of the surface obtained by rotating the curve about the given axis.
Use Simpson's Rule with 
.
(a) 
, 
; 
-axis.
http://matrix.skku.ac.kr/cal-lab/cal-8-2-12.html
, 
; 
-axis
