Download Additional Review Questions for Old Test 3 - Calculus II | MTH 252 and more Exams Calculus in PDF only on Docsity!
Additional Review Questions for MTH 252 Test 3
All integrals may be and should be evaluated on your calculator unless otherwise stated.
- Use the washer method the find the volume of the solid that results spinning the region
outlined in Figure 1 about the line y =โ 4. To earn full credit Figure 1 must be annotated
with all of the labels discussed and illustrated in class.
- Use the shell method to find the volume of the solid created by rotating about the line
y =โ 2 the region enclosed by the parabola
2 y = x and the line y = 4. To earn full credit
Figure 2 must be annotated with all of the labels discussed and illustrated in class.
Figure 1
Figure 2
- Estimate
2 2 1
x 1 dx โ
using 6 trapezoids over intervals of equal width.
- Estimate
5 3 1
x dx
using Simpsonโs Rule with 2 n = 10 intervals of equal width.
- Sketch onto figure 3 a complete graph of the parametric equations indicated in the Figure
caption; then use an appropriate integral to determine the arc-length along the entire curve.
6. Sketch onto Figure 4 the function ( )
2
3 cos 2
y x = โ y y โ. Pay special attention to where you
plot the points where the curve goes off the grid and at each point the curve changes
direction (left to right).
Figure 3
2cos 2 4, 5sin 1 2
x t y t
Figure 4
2
10 cos 1. 2
y x = + y y โ
Additional Review Questions for MTH 252 Test 3 โ Answers
1. ( ) ( ) [ ]
2 (^2 2 )
4
b
a
V ฯ R x r x dx ฯ x x dx ฯ
โ
4
0
d
c
V y h y dy y y dy
x
ฮ = = Define ( )
2 f x = x + 1. Then
(^2 )
1 1 6
x dx T
f f f f f f f
โ
x
ฮ = = Define ( )
3 f x = x. Then
5 3 10 1
1 4 1.4 2 1.8 4 2.2 2 2.6 4 3 2 3.4 4 3.8 2 4.2 4 4.6 5 3 156
x dx S
f f f f f f f f f f f
โ
= โกโฃ + + + + + + + + + + โคโฆโ
=
Figure 3
2 2
2 (^3 2 ) 4sin 2 cos 2 2
b
a
dx dy L dt dt dt
t t dt
ฯ
ฯ
โ โ^ โ โ
Figure 4
1
2
1 2 1 2
2
5 /
5
5 /
2 5
2
2
2
sin sin
5 lim
sin 5 lim^1
lim cos 5
lim cos cos 5 5
cos 0 cos 5 5
t t t t t t t
u x dx dx x x (^) du x dx
x x du dx
du x u
x u
x
u
t
ฯ
ฯ
ฯ
ฯ
โ
โ โ
โ โ
โ โ
โ โ
โ โ
= =^ โ^ =
โ โ^ โ^ โ โ โ
t โ u = 5 ฯ/ t
7. From Hooke's law we know that F ( x ) = kx where x is the displacement of the spring from its
natural length. Since F ( ) 2 = 8 = 2 k , k = 4. Hence the work done while stretching the
spring from its natural length to 4 inches beyond its natural length is given by:
4
0
W xdx
32 in-lb of work is done when stretching the spring from a length of 20 in to a length of 24 in.
- a.
b.
5 5
4 4 4 4 4 3 3
lim lim 2 6 2 6 2 6
t
t t t
dx dx dx
x x x
โ (^) โ โ^ โ โ+
2 6 4 4 4 14 3 3
3 2 6
3 14
3 3 3
lim lim 2 6 2 2
lim (^2) 2 3 4 14
lim (^6 2 6 )
t t
t t
t
t
t
dx du u x u x du^ dx
du u dx
x u
x t u t
t
โ โ
โ
โ
โ โ
โ โ^ โ โ
โ โ
โ โ
โ
= โ^ โ
โ
= โ^ โ โ^ โ โโ
โ โ^ โโ
โ โ^ โ โ
