Calculus I Exam 5 Review (Sections 5.1-5.5)
Review problems from pp. 309-310: 1, 2a, 7 – 15 odd, 21 – 25 odd, 29, 33, 35, 41, 42 plus the
following problems. In addition to the problems listed from the chapter review and these
problems, all of your homework problems are fair game for the test as well. Please be sure to
have your homework completed and ready to hand in on Thursday.
1. By reading values from the graph of f, use four rectangles to find a) a lower estimate and
b) an upper estimate for the area under the graph of 𝑓 from x = 0 to x = 8. Show your work!
2. Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and
values of the rate at two hour time intervals are shown in the table. Find upper and lower
estimates for the total amount of oil that leaked out.
Time (hours)
Rate (Liters/hr)
0
8.8
2
7.9
4
6.7
6
6.4
8
5.6
10
5.3
3. If 𝑓(𝑥) = 5𝑥 2 − 4𝑥, 0 ≤ 𝑥 ≤ 3, evaluate the Riemann Sum for n = 6, taking sample points to
be right endpoints.
4. Use the midpoint rule with the given value of n to approximate the integral. Round answer to
four decimal places. Note: Make sure your calculator is in radian mode!
80
∫ cos √𝑥𝑑𝑥,
0
𝑛=4
5. The graph of f is shown. Evaluate each integral by interpreting it in terms of area.
(Remember that area below the x-axis is counted as negative.)
2
a) ∫0 𝑓(𝑥)𝑑𝑥
5
b) ∫2 𝑓(𝑥)𝑑𝑥
8
c) ∫5 𝑓(𝑥)𝑑𝑥
8
d) ∫0 𝑓(𝑥)𝑑𝑥
Evaluate each integral.
3
6. ∫0 (1 + 6𝑤 2 − 10𝑤 4 )𝑑𝑤
𝜋
7. ∫0 (3𝑒 𝑥 + 6 sin 𝑥)𝑑𝑥
8. The acceleration function of a particle (in 𝑚𝑒𝑡𝑒𝑟𝑠/𝑠𝑒𝑐 2) is 𝑎(𝑡) = 4𝑡 + 2.
a) If the initial velocity is 7 meters/sec, find the velocity function v(t).
b) Find the displacement of the particle from t = 0 to t = 4 seconds.
Use part I of the Fundamental Theorem of Calculus to find the derivative of each function.
𝑥
9. 𝑔(𝑥) = ∫3 𝑒 5𝑡−7 𝑑𝑡
𝑥3
10. 𝑔(𝑥) = ∫1 √𝑠𝑖𝑛 𝑡 𝑐𝑜𝑠 𝑡 5 𝑑𝑡
𝑥
11. Let 𝑔(𝑥) = ∫0 𝑓(𝑡)𝑑𝑡, where f(t) is the function whose graph is shown.
a) Evaluate g(0), g(1), g(2), g(3), g(4), g(5), g(6), g(7), and g(8).
b) Sketch a graph of 𝑔(𝑥).
Find the average value of each function on the given interval.
3
12. 𝑓(𝑥) = 7 √𝑥 , [8, 27]
13. 𝑓(𝑥) = 2𝑥 − 7, [−3, 4]
Evaluate each indefinite integral. (Hint: Use the substitution rule.)
14. ∫
𝑥
(𝑥 2 +8)2
𝑑𝑥
15. ∫ sin(9𝜋𝑡) 𝑑𝑡
16. ∫ 𝑒 𝑥 √43 + 𝑒 𝑥 𝑑𝑥
17. ∫
𝑑𝑥
9−7𝑥
Evaluate each definite integral.
1 𝑒 𝑥 +1
18. ∫0
𝑒 𝑥 +𝑥
𝑑𝑥
9
19. ∫6 √𝑥 − 5𝑑𝑥
1
20. ∫0 𝑥 2 (5𝑥 3 − 4)6 𝑑𝑥