Practice Test 2
(sections 5.9 - 6.5)
1. How many types of improper integrals are there? Give an example of each type.
2. State the Trapezoid Rule, the Midpoint Rule, and Simpson’s Rule. Which of these three
is the most accurate?
𝑇𝑛 =
𝑀𝑛 =
𝑆𝑛 =
3. What is the general formula for the arc length of a function?
4. What is the formula for the average value of a function? Can we always find an x-value
where for that x-value such that 𝑓(𝑥) = 𝑓𝐴𝑉𝐺 ?
5. What are all the important formulas you need to know when finding a centroid?
6. Calculate the following integrals:
a.
∞
1
𝑑𝑥
−∞ 𝑥 2 +1
b.
∞ −5𝑥
𝑒
1
c.
5 3
𝑑𝑥
0 2𝑥−5
d.
1 4
𝑑𝑥
0 𝑥2
𝑑𝑥
7. Calculate the following using numerical integration:
𝜋 /4
0
sin 𝑥 2 𝑑𝑥
a. Which of the following is the error estimate for the Trapezoid Rule?
𝐾 𝑏−𝑎 3
12𝑛 2
𝐾 𝑏−𝑎 3
24𝑛 2
𝐾 𝑏−𝑎 5
180𝑛 4
b. If we want our answer to be accurate to within 0.01 of the true solution, what
should our n be for the trapezoid rule?
c. Using the n you found in part b, find 𝑇𝑛
d. Find the true solution of the integral (either by hand or using your calculator).
e. What is the actual error of the 𝑇𝑛 you found in part c.
8. Find the area between the following curves. Draw a picture of the situation, decide
whether to integrate with respect to x or y, and make sure to find the correct bounds of
the figure. Be careful some of these are trick questions.
a. Find the area between 𝑦 = 4𝑥 + 1 and 𝑦 = 𝑒 𝑥 and 𝑥 = 2
b. Find the area between 𝑦 = 3𝑥 and 𝑦 2 + 8𝑥 = 1
c. Find the area between 𝑦 = sin 𝑥 and 𝑦 = 2sin 𝑥 on the interval [0, 2𝜋]
d. Find the area enclosed by 𝑦 = 𝑥 2 and 𝑦 = 𝑒 𝑥 + 1 and 𝑦 = −𝑥 + 2
9. Suppose each of the given regions is rotated about the given axis of rotation. Find the
volume of the resulting figures.
a. The region defined by the area enclosed by the functions 𝑦 = 𝑥 and 𝑦 = 𝑥/4 is
rotated about the x-axis.
b. The region defined by the area enclosed by the functions 𝑦 = 𝑥 and 𝑦 = 10𝑥 4 is
rotated about 𝑥 = −2
c. The region is defined by the area enclosed by the function 𝑦 = 𝑥 𝑥 − 4 and is
rotated about the y-axis.
10. Consider the straight line 𝑦 = 2𝑥 + 1
a. What is the arc length of this function on the interval [0, 2]?
b. Use the distance formula 𝑑 =
𝑥1 − 𝑥2 2 + 𝑦1 − 𝑦2
2
to find the distance
between the points (0, 1) and (2, 5).
c. Do you get the answer you expect? Why or why not?
11. Consider the parametric function: 𝑥 𝑡 = 5 cos 𝑡 and 𝑦 𝑡 = 5 sin 𝑡
a. What is the arc length of this function on 0 ≤ 𝑡 ≤ 𝜋 (keep your answer in terms of 𝜋)?
b. Graph this function. (You should get a circle.)
c. Use the formula for the circumference of a circle (𝐶 = 2𝜋𝑟) to find the
circumference of one half of a circle with radius 5. Do you get the answer you
expect?
12. Consider the function 𝑓 𝑥 = 𝑥 + 2
a. Find the Mean value of 𝑓 𝑥
2
on the interval [0,9]
b. Find all values 𝑐 where 0 ≤ 𝑐 ≤ 9 and 𝑓(𝑐) = 𝑓𝐴𝑉𝐺
6m
13. A trough has two ends which are isosceles trapezoids with bases
6 meters and 4 meters, a height of 5 meters and a length of 8
meters (see diagram). Suppose this trough is filled with water
(with density 1000 kg/m3). Find the work required to pump the
water out of the tank, leaving a depth of 1 meter at the bottom of
the trough.
5m
8m
4m
14. A force of 25 pounds is required to compress a spring 6 inches from its natural length of
30 inches. Find the work done in compressing the spring an additional 6 inches.
15. For each of the following situations, find the moments & the Centers of Mass (Centroids)
a. (0, 5), (1, -4), (2, 3), (-1, 0) with the associated masses 3, 5, -3, 1
b. The region between the curve 𝑦 = −2𝑥(𝑥 − 2) and the x-axis
c. The region between the two curves 𝑦 = 𝑥 and 𝑦 = 𝑥 3 where 𝑥 ≥ 0