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MATH 152 { Section C.1
Review Sheet 3
Sections 9.1-9.2, 8.2, 9.3-9.5
Answers are not guaranteed, but if you nd an incorrect answer, please let me know.
1. Eliminate the parameter and sketch the curve
x = et y = 4e2t:
30
25
20
15
10
5
0
0.511.522.5
Answer: y = 4x2,x > 0
2. Eliminate the parameter and sketch the curve
x = 5 cos t y = 3 sin t:
3
2
1
0 24
-4-2
-1
-2
-3
y 2
+
5
3 =1
x 2
3. Eliminate the parameter and sketch the curve
x = sec t y = tan t:
1
0.5
00.51
-1
-0.5
-0.5
-1
x2 , y2 = 1
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Review Sheet 3
4. (a) First write the equation of the line tangent to the given parametric curve at the point
corresponding to the given value of t. (b) Then calculate d2 y=dx2 in order to determine
whether the curve is concave upward or concave downward at this point
x = t sin t y = t cos t; t = =2:
2
(a) y = 4 , 2 x; (b) concave downward
5. The curve C is determined by the parametric equations x = e,t and y = e2t . Calculate
dy=dx and d2y=dx2 directly from these parametric equations. Conclude C is concave upward
at every point.
2.52
1.51
0.5
0.6
0.811.2
1.4
1.6
,2e3t ; 6e4t; y = x,2, x > 0
6. Parametrize the parabola y 2 = 4px by expressing x and y as functions of the slope m of the
tangent line at the point P (x; y ) of the parabola.
p
y = 2p
m , x = m2
7. Set up and simplify the integral that gives the length of the given smooth arc; do not evaluate
the integral.
y = 1 , x2 ; 0 x 100:
Z 100 p
p
p
1 + 4x2 dx = 50 40001 + 41 ln 200 + 40001 10001:6
0
For problems 8-9, nd the lengths of the arcs.
8. y = 23 (x2 + 1)3=2, 0 x 2.
Z 2p
1 + (x2 + 1)(4x2) dx = 22
3
0
9. y = 61 x3 + 21x , 1 x 3.
Z 3
1
s
2
1 + 21 x2 , 2x1 2 dx = 14
3
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Review Sheet 3
For problems 10-11, nd the area of the region between the parametric curve and the x-axis.
10. x = cos t, y = sin2 t, 0 t .
Z 0
11. x = cos t, y = et , 0 t .
Z 0
sin2 t(, sin t) dt = 34
et (, sin t) dt = 21 e + 12
p
12. Compute the area of the loop formed by x = 3t2 , y = 3t , 13 t3 . (The loop is generated by
,3 t 3.
!
p
Z 3
Z ,3 Z 3
p
p
p 2 2p3 4
1
1
216
3
3
3t , 3 t 2 3t dt ,
3t , 3 t 2 3t dt =
6 3t , 3 t dt = 5 3
0
,3
0
For problems 13-14, nd the arc length of the given curve.
13. x = 2t, y = 32 t3=2 , 5 t 12.
Z 12
5
14. x = sin t , cos t, y = sin t + cos t, 4 t 2 .
p
4 + t dt = 74
3
p
2
4
For problems 15-16, nd the area of the surface of revolution generated by revolving the given
curve around the indicated axis.
p
15. x = 1 , t, y = 2 t, 1 t 4, x-axis.
Z 4
1
16. x = 2t + 3, y = t3 , ,1 t 1, x-axis.
p
q
,
3=2
3=2
2(2 t) (,1)2 + t,1=2 2 dt = 8
3 5 ,2
Z 1
,1
h
q
i
3=2
2 t3 (2)2 + (3t2 )2 dt = 2
27 13 , 8
h
i
For problems 17-18, nd dy=dx in terms of . In 17, nd all for which the tangent to the
curve is horizontal or vertical.
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Review Sheet 3
p3
17. r = e
.
p
p3sin + cos ; H: = , 3 + k; V: = 3 + k
3cos , sin 18. r = sin(3).
sin(3) cos + 3 cos(3) sin , sin(3) sin + 3 cos(3) cos 19. Find the area of the surface of revolution generated by revolving r = e , 0 2 around
the x-axis.
p
Z
q
=2
0
,
2 e sin (e )2 + (e )2 d = 2 52 (1 + 2e )
p
20. Find the area of the surface of revolution generated by revolving x = 2t2 + 1t , y = 8 t,
1 t 2 around the x-axis.
s
2 2
Z 2
i
h p
p 1
1
28 t 4t , t2 + 4 p dt = 16
2
+
2
11
5
t
1
21. Compute the arc length of the following parametric curve: x = 1t , t12 , y = t14 , 1 t T .
Z T
1
s
, t12 + t23
2
+ , t45
2
dt = 2 , T1 , T12
22. Take the limit in problem 21 as T ! 1. Note that the arc length of the parametric curve
x = 1t , t12 , y = t14 , t 2 [1; 1) is nite even though the interval is innite. Sketch the curve.
1
0.8
0.6
0.4
0.2
0
0.05 0.1 0.15 0.2 0.25
Limit= 2.
4