# Math 1172 Sample Test #1 + 1 and the line

```Math 1172 Sample Test #1
1.
Find the area of the region enclosed by the curve y = x 2 + 1 and the line y = 2x + 1
€
2.
€
Find the area of the region enclosed by the graphs of x = y 2 and y = x − 2
3.
2.
€
€
1.
-1.
1.
2.
3.
4.
5.
-1.
-2.
-3.
3.
Find the length of the curve y =
x 4 x −2
+
, 1≤ x ≤ 2
8
4
⎛ dy ⎞ 2
(Hint: 1+ ⎜ ⎟ is a perfect square.)
⎝ dx ⎠
4.
€
Find the length of the curve y = 2(x + 1) 3 / 2 , 0 ≤ x ≤ 1
€
5. Find the area of the surface obtained by
x 4 x −2
+
, 1 ≤ x ≤ 2 about the x-axis.
(a)
rotating the €
curve y =
8
4
x 4 x −2
+
, 1 ≤ x ≤ 2 about the y-axis.
(b) rotating the curve y =
8
4
€
€
Math 1172 Sample Test #1
Page 2
Use this sketch for problems 6 and 7.
6.
Use the washer method to find the volume of the solid generated by rotating the
region bounded by the curve y = x2 and the line y = x about the x-axis.
7.
Use the cylindrical shell method to find the volume of the solid generated by rotating
the region bounded by the curve y = x2 and the line y = x about the y-axis.
8.
Find the volume of the solid generated by rotating the region bounded by the curve
x = y2 + 1 and the line x = 2y + 1 about the x-axis.
9.
A spring has a natural length of 0.3m. It takes a force of 10N to keep the spring
stretched to a length of 0.35m. Find the work done in stretching the spring from its
natural length to a length of 0.4m.
Math 1172 Sample Test #1
Page 3
For problems 10 and 11, use ρ = 1000 kg/m3 and g = 9.8 m/s2
10. A cylindrical tank of radius 2 meters and height 5 meters is positioned on a tower so
that the bottom of the tank is 20 meters above the ground. Find the work done to fill
€ pumped up from
€ ground level.
the tank with water
11. The lower edge of a dam is defined by the parabola y = x 2 /4, −10 ≤ x ≤ 10 . The
upper edge of the dam is defined by the line y = 25 . (Lengths are measured in
meters.) Find the force on the dam if the water level is at the top of the dam.
€
12. The population of a town increases from 30,000
in 1990 to 40,000 in 2005. Devise
€
the exponential growth function that fits this data and use it to predict when the
population will reach 50,000.
13. A drug is eliminated from a body at a rate of 12% per hour. Devise the exponential
growth function that fits this data and use it to predict when the amount of the drug
will reach 20% of the initial dose.
14. Uranium-238 (U-238) has a half-life of 4.5 billion years. 75% of the original U-238
in a rock still remains. How old is the rock?
15. Find the indefinite integrals:
dx
(a)
∫ 1 − sin x
(b)
∫
(c)
∫ sec(2x)tan(2x)dx
(d)
∫x
(e)
∫e
(f)
∫x2
(g)
∫x
€
€
€
€
x 2 +1
dx
x +1
x +1
dx
2
+1
5x
€
€
€
2
dx
x2
dx
dx
− 6x + 25
Math 1172 Sample Test #1
Page 4
16. Evaluate the definite integrals.
(a)
∫
(b)
∫
€
4
x
0
2
x +9
dx
1
0
dx
4 − x2
17. Use integration by parts to find the integrals.
€ (a)
€
€
€
€
€
€
€
∫ x e dx
2 x
(b)
∫x
(c)
∫ xe
5x
(d)
∫x
2
(e)
∫ x cos 3x dx
(f)
∫ ln x dx
(g)
∫ arctan x dx
2
ln x dx
dx
cos x dx
```