3.2. ARC LENGTH AND SURFACE AREA 95
PROBLEMS
Problem 3.25 For each of the following functions, compute ds, the differential of arc length.
1. f .x/ D x
3
2. g.x/ D sin.x/
3. h.x/ D tan
1
.x/
4. r.x/ D
x
x C 1
5. s.x/ D e
x
6. q.x/ D x
3=4
7. a .x/ D
1
3
8. b.x/ D 2
x
Problem 3.26 For each of the following functions, compute the arc length of the graph of the
function on the given interval.
1. f .x/ D 9x
2=3
I Œ0; 1
2. g.x/ D 2x C 1I Œ0; 4
3. h.x/ D 2x
3=2
I Œ2; 5
4. r.x/ D x
2
C 4x C4I Œ0; 6
5. s.x/ D
p
.x 2/
3
I Œ4; 5
6. q.x/ D .x C 1/
2=3
I Œ1; 1
7. a .x/ D 4x
3=2
I Œ2; 4
8. b.x/ D x
2
C x C 1=4I Œ0; 8
Problem 3.27 For each of the following functions, compute the surface area of rotation of the
function for the given interval.
1. f .x/ D xI Œ0; 3
2. g.x/ D
p
xI Œ4; 9
3. h.x/ D e
x
I Œ0; 1
4. r.x/ D sin.x/I Œ0;
5. s.x/ D cos.3x/I Œ0; =4
6. q.x/ D sin.x/ cos.x/I Œ0; =2
96 3. ADVANCED INTEGRATION
Problem 3.28 Using the techniques for surface area of revolution, find the formula for the
surface area of a cone with apex angle , as shown above. Don’t forget the area of the bottom.
Problem 3.29 If
y D x
2=3
;
find the formula for the arc length of the graph of this function on the interval Œa; b .
Problem 3.30 Based on the material in this section, if
.
f .t/; g.t/
/
is a parametric curve, what would the differential of arc length, ds, be?
Problem 3.31 Derive the polar differential of arc length.
Problem 3.32 Derive the parametric differential of arc length for
.x.t/; y.t //
Problem 3.33 Find but do not evaluate the integral for computing the arc length of
y D sin.x/:
Discuss: what techniques might work for this integral.
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