View
6
Download
0
Category
Preview:
Citation preview
Pre-Calc Intro to Integrals KEY ~1~ NJCTL.org
Riemann Sums β Class Work
1. Consider the region between π¦ = 9π₯ β π₯3and the x-axis for 0 β€ π₯ β€ 3.
a. Sketch the graph of the region partitioned into 6 rectangles with LRAM
b. Calculate the area using LRAM
2. Using the same region as question 1, follow the same step to find RRAM.
3. Using the same region as question 1, follow the same step to find MRAM.
4. Using the same region as question 1, find LRAM, RRAM, & MRAM but with 12 partitions.
5. Make a conjecture about the area of the region in question 1.
ππ. ππππ
ππ. ππππ
ππ. πππππ
LRAM: ππ. ππππππ
RRAM: ππ. ππππππ
MRAM: ππ. πππππππ
About ππ. ππ
Pre-Calc Intro to Integrals KEY ~2~ NJCTL.org
Find LRAM, RRAM, and MRAM between f(x) and the x-axis. Given are the bounds [a,b] and the
number of partitions n.
6. π(π₯) = βπ₯, [0,10], π = 5
7. π(π₯) = π₯3, [1,3], π = 4
8. π(π₯) = cos π₯, [0,2π], π = 8
The table shows the rate of fuel consumption of a car at given times on a 2 hour trip.
Time 10am 10:15 10:30 10:45 11am 11:15 11:30 11:45 Noon
gal/hour 2 3 3 4 3 2 2 3 4
9. Using 4 partitions and MRAM, estimate the area.
10. What does this area represent?
11. What are the appropriate units for the area?
LRAM: ππ. πππ
RRAM: ππ. ππππ
MRAM: ππ. πππ
LRAM: ππ
RRAM: ππ
MRAM: ππ. ππ
LRAM: π
RRAM: π
MRAM: π
MRAM: π
Approximate number of gallons consumed
Gallons
Pre-Calc Intro to Integrals KEY ~3~ NJCTL.org
Reimann Sums β Homework
12. Consider the region between π¦ = 3π₯ β π₯2and the x-axis for 0 β€ π₯ β€ 3.
a. Sketch the graph of the region and partition into 6 rectangles with LRAM
b. Calculate the area using LRAM
13. Using the same region as question 12, follow the same step to find RRAM.
14. Using the same region as question 12, follow the same step to find MRAM.
15. Using the same region as question 12, find LRAM, RRAM, & MRAM but with 12 partitions.
16. Make a conjecture about the area of the region in question 12.
π. πππ
π. πππ
π. πππ
LRAM: π. πππππ
RRAM: π. πππππ
MRAM: π. ππππππ
About π. π
Pre-Calc Intro to Integrals KEY ~4~ NJCTL.org
Find LRAM, RRAM, and MRAM between f(x) and the x-axis. Given are the bounds [a,b] and the
number of partitions n.
17. π(π₯) = π₯2, [1,9], π = 4
18. π(π₯) = βπ₯3 [0,8], π = 4
19. π(π₯) = sin π₯, [0, π], π = 4
The table shows the rate of downloads of a new song in the first 6 hours it was available.
Time 12am 12:45 1:30 2:15 3:00 3:45 4:30 5:15 6am
downloads/min 200 100 90 80 50 20 25 35 24
20. Using 4 partitions and MRAM, estimate the area.
21. What does this area represent?
22. What are the appropriate units for the area?
LRAM: πππ
RRAM: πππ
MRAM: πππ
LRAM: π. πππ
RRAM: ππ. πππ
MRAM: ππ. ππ
LRAM: π. πππ
RRAM: π. πππ
MRAM: π. πππ
MRAM: πππππ
Approximate number of downloads
Downloads
Pre-Calc Intro to Integrals KEY ~5~ NJCTL.org
Trapezoid Rule β Class Work
23. Consider the region between π¦ = 4π₯ β π₯3and the x-axis for 0 β€ π₯ β€ 2.
a. Sketch the graph of the region and partition into 4 trapezoids.
b. Calculate the area.
24. Using the same region as question 23, apply the trapezoid rule but with 8 partitions.
25. Make a conjecture about the area of the region in question 23.
Find the area using the trapezoid rule between f(x) and the x-axis. Given are the bounds [a,b]
and the number of partitions n.
26. π(π₯) =1
π₯, [2,4], π = 4
27. π(π₯) = π₯ β π₯3, [1,3], π = 4
28. π(π₯) = sin π₯, [0,2π], π = 6
The table shows the speed of a car at given times on a 2 hour trip.
Time 10am 10:15 10:30 10:45 11am 11:15 11:30 11:45 Noon
miles/hour 65 50 60 45 70 60 55 60 0
29. Using 4 partitions and trapezoid rule to estimate the area.
30. What does this area represent?
31. What are the appropriate units for the area?
π. ππ
π. ππππ
About π
π. πππ
βππ. π
π
πππ. ππ
Approximate number of miles driven
Miles
Pre-Calc Intro to Integrals KEY ~6~ NJCTL.org
Trapezoid Rule β Homework
32. Consider the region between π¦ = 8 β 4π₯ and the x-axis for 0 β€ π₯ β€ 2.
a. Sketch the graph of the region and partition into 4 trapezoids.
b. Calculate the area using the trapezoid rule
33. Using the same region as question 32, apply the trapezoid rule but with 10 partitions.
34. Make a conjecture about the area of the region in question 32.
Find the area using the trapezoid rule between f(x) and the x-axis. Given are the bounds [a,b]
and the number of partitions n.
35. π(π₯) = π₯2 β 4, [1,3], π = 6
36. π(π₯) = β4 β π₯[0,4], π = 4
37. π(π₯) = cos π₯, [0, π], π = 6
The table shows the rate of typists typing a manuscript over a 6 hour period.
Time Noon 12:45 1:30 2:15 3 3:45 4:30 5:15 6pm
words/min 200 100 90 80 50 20 25 35 24
38. Using 4 partitions and trapezoid rule to estimate the area.
39. What does this area represent?
40. What are the appropriate units for the area?
π
π
Exactly π
ππ
ππ
π. πππ
π
πππππ
Approximate number of words typed
Words
Pre-Calc Intro to Integrals KEY ~7~ NJCTL.org
Accumulation Functions β Class Work
Use the graph of πβ²(π₯) to answer the following questions. π(0) = 2
41. β« πβ²(π₯)ππ₯3
0
42. β« πβ²(π₯)ππ₯5
0
43. β« πβ²(π₯)ππ₯0
β4
44. β« πβ²(π₯)ππ₯0
3
45. β« πβ²(π₯)ππ₯β4
5
46. πβ²(2)
47. π(2)
48. π"(2)
49. When is π"(π₯) > 0?
π
π. π
π
βπ
βπ. π
π
π
π
[βπ, βπ]πππ [π, π]
Pre-Calc Intro to Integrals KEY ~8~ NJCTL.org
Homework
Use the graph of πβ²(π₯), a semi-circle and two lines to answer the following questions. π(1) = 0
50. β« πβ²(π₯)ππ₯3
0
51. β« πβ²(π₯)ππ₯5
0
52. β« πβ²(π₯)ππ₯0
β4
53. β« πβ²(π₯)ππ₯β4
0
54. β« πβ²(π₯)ππ₯β4
5
55. πβ²(3)
56. π(4)
57. π"(β2)
58. When is π(π₯) increasing?
π. π
π. π
π. ππ
βπ. ππ
βππ. ππ
βπ
π
π
[π, π]
Pre-Calc Intro to Integrals KEY ~9~ NJCTL.org
Anti-Derivatives β Class Work
β« π(π₯)ππ₯ = 42
β2
, β« π(π₯)ππ₯ = β32
β2
, β« π(π₯)ππ₯ = 8,5
2
β« π(π₯)ππ₯ = 32
0
, β« π(π₯)ππ₯ = 25
8
59. β« (π(π₯) + π(π₯))ππ₯2
β2 60. β« (π(π₯) β π(π₯))ππ₯
2
β2 61. β« (3π(π₯) + |π(π₯)|)ππ₯
2
β2
62. β« π(π₯)ππ₯5
β2 63. β« 4π(π₯)ππ₯
8
2 64. β« π(π₯)ππ₯
8
β2 65. β« π(π₯)ππ₯
0
β2
Find the value of following definite integrals.
66. β« 3ππ₯4
1 67. β« π₯ππ₯
5
2 68. β« 4π₯3ππ₯
3
β2
69. β«1
π₯
5
1ππ₯ 70. β« ππ₯ππ₯
6
0 71. β« (3π₯2 + 6π₯ β 5)ππ₯
1
β2
72. β«1
π₯2 ππ₯2
1 73. β« π ππ2π₯
π
40
ππ₯ 74. β« sin π₯ ππ₯2π
0
75. β«1
1+π₯2 ππ₯1
0
π π ππ
ππ ππ ππ π
π ππ. π ππ
π₯π§ π β π. πππ ππ β π β πππ. ππ βππ
π
π π π
π
π
Pre-Calc Intro to Integrals KEY ~10~ NJCTL.org
Anti-Derivatives β Homework
β« π(π₯)ππ₯ = 52
β2
, β« π(π₯)ππ₯ = 92
β2
, β« π(π₯)ππ₯ = β6,5
2
β« π(π₯)ππ₯ = β12
0
, β« π(π₯)ππ₯ = β75
8
76. β« (π(π₯) + 2π(π₯))ππ₯2
β2 77. β« (2π(π₯) β 2π(π₯))ππ₯
2
β2 78. β« (3π(π₯) + |π(π₯)|)ππ₯
2
β2
79. β« (π(π₯) + 1)ππ₯5
2 80. β« 3π(π₯)ππ₯
8
2 81. β« π(π₯)ππ₯
8
β2 82. β« π(π₯)ππ₯
0
β2
Find the value of following definite integrals.
83. β« 4ππ₯4
2 84. β« 2π₯ππ₯
5
2 85. β« π₯3ππ₯
6
β1
86. β«2
π₯
6
1ππ₯ 87. β« (4ππ₯ + 1)ππ₯
5
0 88. β« (3π₯2 + 6π₯ β 5)ππ₯
1
β2
89. β«6
π₯3 ππ₯2
1 90. β« π πππ₯π‘πππ₯
π
40
ππ₯ 91. β« cos π₯ ππ₯2π
0
92. β«1
β1βπ₯2ππ₯
1
20
ππ βπ ππ
βπ π π π
π ππ πππ. ππ
π π₯π§ π β π. ππ πππ + π β πππ. ππ π
π. ππ βπ β π β π. πππ π
π
π
Pre-Calc Intro to Integrals KEY ~11~ NJCTL.org
Fundamental Theorem of Calculus β Class Work
Find ππ¦
ππ₯
93. π¦ = β« (4π‘ β 2)ππ‘π₯
1 94. π¦ = β« (3π’2 β 4π’)ππ’
2π₯
2
95. π¦ = β« ππ(π£)ππ£4
π₯ 96. π¦ = β« (4π‘3 β 2π‘)ππ‘
0
π₯2
97. π¦ = β« (7π’)ππ’2π₯
3π₯2 98. π¦ = β« ππ£ππ£π₯ ππ π₯
ππ π₯
99. Let πΉ(π₯) = β« π(π‘)ππ‘,π₯
0 where π(π‘) is defined by the graph.
a. π(2)
b. πΉ(2)
c. πΉβ²(2)
d. πβ²(2)
100. β« (5π’ β 6)ππ’ + πΎπ₯
β2= β« (5π’ β 6)ππ’
π₯
4, find K
π π
π π= ππ β π
π π
π π= ππππ β πππ
π π
π π= β π₯π§ π
π π
π π= βπππ + πππ
π π
π π= πππ β πππππ π π
π π= π₯π§ π ππ π₯π§ π + ππ π₯π§ π β
ππ₯π§ π
π
π(π) = π
π(π) = π
πβ²(π) = π
πβ²(π) = βπ
π
π² = π
Pre-Calc Intro to Integrals KEY ~12~ NJCTL.org
Fundamental Theorem of Calculus β Homework
Find ππ¦
ππ₯
101. π¦ = β« ππ’ππ’π₯
2 102. π¦ = β« π‘2ππ‘
βπ₯
3
103. π¦ = β« π ππ π£ ππ£π
π₯ 104. π¦ = β« β5 β π‘ ππ‘
5
4βπ₯
105. π¦ = β« (π’2 β 4π’ + 2)ππ’7π₯
2π₯ 106. π¦ = β« βπ£
π₯2
1
π₯
ππ£
107. Let πΉ(π₯) = β« π(π‘)ππ‘,π₯
0 where π(π‘) is defined by the graph.
a. π(2)
b. πΉ(2)
c. πΉβ²(2)
d. πβ²(2)
108. β« (3π’2 + 2π’ + 1)ππ’ + πΎπ₯
1= β« (3π’2 + 2π’ + 1)ππ’
π₯
3, find K
π π
π π= ππ π π
π π=
βπ
π
π π
π π= β π¬π’π§ π
π π
π π= βπ + π
π π
π π= πππππ β ππππ + ππ π π
π π= πππ +
βπ
ππ
π(π) = π
π(π) = π
πβ²(π) = π
πβ²(π) = πππ ππππππ
π² = βππ
Pre-Calc Intro to Integrals KEY ~13~ NJCTL.org
Substitution Method β Class Work
Evaluate the indefinite integral using the Substitution Method
109. β« 2π₯βπ₯2 + 1ππ₯ 110. β«π₯3
(π₯4+1)4 ππ₯
111. β« sin5 π₯ cos π₯ ππ₯ 112. β« π₯ cos(π₯2) ππ₯
113. β« π₯πβπ₯2ππ₯ 114. β«
ππ₯
π₯ ln π₯
Evaluate the definite integral
115. β«π₯
(π₯2+1)3 ππ₯1
0 116. β« (π₯ + 1)(π₯2 + 2π₯)3ππ₯
2
1
117. β« π₯ tan(π₯2) ππ₯1
0 118. β« β5π₯ + 6
2
β1ππ₯
π
π(ππ + π)π/π + πͺ β
π
ππ(ππ+π)π + πͺ
π
ππ¬π’π§π π + πͺ
π
ππ¬π’π§(ππ) + πͺ
βπ
ππβππ
+ πͺ π₯π§(π₯π§ π) + πͺ
π
ππ
ππππ
π= πππ. πππ
π
π(πβπ β π) β π. πππ π. π
Pre-Calc Intro to Integrals KEY ~14~ NJCTL.org
Substitution Method β Homework
Evaluate the indefinite integral using the Substitution Method
119. β« π₯2(π₯3 + 1)4ππ₯ 120. β«1
(π₯+2)2 ππ₯
121. β«2π₯2+π₯
(4π₯3+3π₯2)5 ππ₯ 122. β« sin(2π₯ β 4) ππ₯
123. β«π₯
βπ₯2+9ππ₯ 124. β« sec2 π₯ (4 tan3 π₯ β 3 tan2 π₯)ππ₯
Evaluate the definite integral
125. β« π₯βπ₯2 + 94
0ππ₯ 126. β«
π₯+3
(π₯2+6π₯+1)3 ππ₯2
0
127. β« (π₯ β 9)β2/3ππ₯17
10 128. β« tan2 π₯ sec2 π₯ ππ₯
π/4
0
π
ππ(ππ + π)π + πͺ β
π
(π+π)+ πͺ
βπ
ππ(πππ+πππ)π + πͺ β
π
πππ¨π¬(ππ β π) + πͺ
βππ + π + πͺ πππ§π π β πππ§π π + πͺ
ππ
πβ ππ. ππ
ππ
πππβ π. πππ
π π
π
Pre-Calc Intro to Integrals KEY ~15~ NJCTL.org
Area Between Curves β Class Work
Find the total area between the functions.
129. π¦ = π₯4 β 6π₯2 πππ π¦ = 6 β π₯2 130. π¦ = cos π₯ πππ π¦ = 2cos π₯ πππ 0 β€ π₯ β€π
2
131. π¦ = π₯3 β 3π₯ πππ π¦ = π₯ πππ π₯ β₯ 0 132. π₯ = π¦2 β 6π¦ πππ π₯ + π¦ = 6
133. π₯ = π¦2 πππ π₯ = π¦ 134. π¦ = cos π₯ πππ π¦ = π₯2 β π2
Area Between Curves β Homework
Find the total area between the functions.
135. π¦ = 2 + 3π₯2 πππ π¦ = 6 β 2π₯2 136. π¦ = sin π₯ πππ π¦ = cos π₯ πππ 0 β€ π₯ β€ π
137. π¦ = β2π₯3 + 10π₯ πππ π¦ = β8π₯ πππ π₯ β€ 0 138. π₯ = βπ¦2 + 9 πππ π₯ + π¦ = 3
139. π₯ = π¦2 πππ π₯ = π¦2
3β 140. π¦ = sin π₯ πππ π¦ = π₯2 β ππ₯
ππ
πβπ β ππ. πππ π
π πππ
πβ ππ. ππ
π
π ππ. π
ππ
πβ π. ππ πβπ β π. πππ
ππ. π πππ
πβ ππ. ππ
π
ππβ π. πππ π +
π
ππ π β π. ππ
Pre-Calc Intro to Integrals KEY ~16~ NJCTL.org
Volume: Disk Method β Class Work
Find the volume of the solid.
141. π¦ = 6π₯ β π₯2 revolved about the x-axis
142. Area between π¦ = 6π₯ β π₯2 and the x-axis revolved about π¦ = β5
143. π¦ = π₯, π¦ = β2π₯ + 4, πππ π¦ = 0 revolved about the x-axis
144. π¦ = π₯, π¦ = β2π₯ + 4, πππ π₯ = 0 revolved about the y-axis
π β« (ππ β ππ)ππ ππ
π
ππππ
ππ β πππ. π
π β« (π + ππ β ππ)ππ ππ
π
ππππ
ππ β ππππ. ππ
π β« (π)ππ ππ/π
π+ π β« (π β ππ)ππ π
π
π/π
ππ
πππ β π. πππ
π β« (π)ππ ππ/π
π+ π β« (π β
π
ππ)
π
π ππ
π/π
ππ
πππ β π. ππ
Pre-Calc Intro to Integrals KEY ~17~ NJCTL.org
145. π¦ = 2π₯, π₯ = 3, πππ π₯ β ππ₯ππ revolved about the x-axis
146. π¦ = 2π₯, π₯ = 3, πππ π₯ β ππ₯ππ revolved about the π₯ = 3
147. π¦ = 6 β π₯, π¦ = π₯2, πππ π¦ = 0 revolved about the x-axis
148. π¦ = 6 β π₯, π¦ = π₯2, πππ π₯ = 0 revolved about the y-axis
π β« (ππ)ππ ππ
π
πππ β πππ. πππ
π β« (π βπ
ππ)
π
π ππ
π
πππ β ππ. πππ
π β« (ππ)ππ ππ
π+ π β« (π β π)ππ π
π
π
πππ
πππ β ππ. πππ
π β« (βπ)π
π ππ
π+ π β« (π β π)ππ π
π
π
πππ
ππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~18~ NJCTL.org
Volume: Disk Method β Homework
Find the volume of the solid.
149. π¦ = 8π₯ β 2π₯2 revolved about the x-axis
150. π¦ = 8π₯ β 2π₯2 and the x-axis revolved about π¦ = β3
151. π¦ = 3π₯, π¦ = βπ₯2 + 4, πππ π¦ = 0 revolved about the x-axis
152. π¦ = 3π₯, π¦ = βπ₯2 + 4, πππ π₯ = 0 revolved about the y-axis
π β« (ππ β πππ)ππ ππ
π
ππππ
πππ β πππ. πππ
π β« (π + ππ β πππ)ππ ππ
π
ππππ
πππ β πππ. πππ
π β« (ππ)ππ ππ
π+ π β« (π β ππ)ππ π
π
π
ππ
πππ β ππ. πππ
π β« (π
ππ)
π
π ππ
π+ π β« (βπ β π)
ππ π
π
π
π
ππ β π. πππ
Pre-Calc Intro to Integrals KEY ~19~ NJCTL.org
153. π¦ = 3π₯, π₯ = 4, πππ π₯ β ππ₯ππ revolved about the x-axis
154. π¦ = 3π₯, π₯ = 4, πππ π₯ β ππ₯ππ revolved about the π₯ = 4
155. π¦ = π₯2 + 2, π¦ = 10 β π₯2, π¦ = 0, π₯ = 0, πππ π₯ = 3 revolved about the x-axis
156. π¦ = π₯2 + 2, π¦ = 10 β π₯2, π₯ = 0, πππ π₯ = 3 revolved about the x-axis
π β« (ππ)ππ ππ
π
ππππ β πππ. πππ
π β« (π βπ
ππ)
π
π πππ
π
πππ β πππ. ππ
π β« (ππ + π)ππ ππ
π+ π β« (ππ β ππ)ππ π
π
π
πππ
ππ β πππ. ππ
π β« ((ππ β ππ) β (ππ + π))π
π ππ
π+ π β« ((ππ + π) β (ππ β ππ))
π
π ππ
π
πππ
ππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~20~ NJCTL.org
Volume: Washer Method β Class Work
Find the solid created by rotating the region π¦ = π₯2, π¦ = π₯, π₯ = 1, πππ π₯ = 4 about
157. π¦ = 0
158. π¦ = 1
159. π¦ = 100
160. π¦ = β2
161. π₯ = 0
π β« (ππ)π β (π)ππ ππ
π
πππ
ππ β πππ. πππ
π β« (ππ β π)π β (π β π)ππ ππ
π
πππ
ππ β πππ. πππ
π β« (πππ β π)π β (πππ β ππ)ππ ππ
π
πππππ
ππ β ππππ. πππ
π β« (π + ππ)π β (π + π)ππ ππ
π
ππππ
ππ β πππ. πππ
π β« (π)π β (βπ)π
π ππ
π+ π β« (π)π β (βπ)
ππ π
ππ
π
πππ
ππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~21~ NJCTL.org
162. π₯ = β2
163. π₯ = 1
164. π₯ = 4
165. π₯ = 10
π β« (π + π)π β (π + βπ)π
π ππ
π+ π β« (π)π β (π + βπ)
ππ π
ππ
π
πππ
ππ β πππ. πππ
π β« (π β π)π β (βπ β π)π
π ππ
π+ π β« (π)π β (βπ β π)
ππ π
ππ
π
πππ
ππ β πππ. πππ
π β« (π β βπ)π
β (π β π)ππ ππ
π+ π β« (π β βπ)
ππ π
ππ
π
ππ
ππ β ππ. πππ
π β« (ππ β βπ)π
β (ππ β π)ππ ππ
π+ π β« (ππ β βπ)
πβ (π)ππ π
ππ
π
πππ
ππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~22~ NJCTL.org
Volume: Washer Method β Homework
Find the solid created by rotating the region π¦ = βπ₯3, π¦ = βπ₯, π₯ = 1, πππ π₯ = 3 about
166. π¦ = 0
167. π¦ = 10
168. π¦ = β1
169. π¦ = β50
170. π₯ = 0
π β« (βππ)π β (βπ)ππ ππ
π
ππππ
πππ β πππ. πππ
π β« (ππ + ππ)π β (ππ + π)ππ ππ
π
πππππ
πππ β ππππ. πππ
π β« (π β ππ)π β (π β π)ππ ππ
π
ππππ
πππ β πππ. πππ
π β« (ππ β π)π β (ππ β ππ)ππ ππ
π
πππππ
πππ β ππππ. πππ
π β« (π)π β (ππ/π)π
π ππ
π+ π β« (π)π β (ππ/π)
ππ π
ππ
π
ππππ
πππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~23~ NJCTL.org
171. π₯ = β2
172. π₯ = 1
173. π₯ = 3
174. π₯ = 10
π β« (π + π)π β (π + ππ/π)π
π ππ
π+ π β« (π)π β (π + ππ/π)
ππ π
ππ
π
ππππ
πππ β πππ. πππ
π β« (π β π)π β (ππ/π β π)π
π ππ
π+ π β« (π)π β (ππ/π β π)
ππ π
ππ
π
πππ
πππ β πππ. πππ
π β« (π β ππ/π)π
β (π β π)ππ ππ
π+ π β« (π β ππ/π)
ππ π
ππ
π
πππ
πππ β ππ. πππ
π β« (ππ β ππ/π)π
β (ππ β π)ππ ππ
π+ π β« (ππ β ππ/π)
πβ (π)ππ π
ππ
π
ππππ
πππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~24~ NJCTL.org
Volume: Shell Method β Class Work
Use the Shell Method to calculate the volume of the object created by rotating the described
region about the given axis.
175. π¦ = 1 + π₯2, π¦ = 0, π₯ = 1, π₯ = 3 revolved about the y-axis
176. π¦ = 8 β π₯3, π¦ = 8 β 4π₯ revolved about the y-axis
177. π¦ = π₯β4, π¦ = 0, π₯ = β3, π₯ = β1 revolved about π₯ = 4
178. π¦ = π₯2, π¦ = 8 β π₯2, π₯ = 0 revolved about π₯ = β3
ππ β« π(π + ππ)π ππ
π
πππ β πππ. πππ
ππ β« π ((π β ππ) β (π β ππ)) π ππ
π
πππ
πππ β ππ. πππ
ππ β« (βπ + π)(πβπ)π πβπ
βπ
πππ
πππ β ππ. ππ
ππ β« (π + π) ((π β ππ) β (ππ)) π ππ
π
πππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~25~ NJCTL.org
179. π¦ = π₯, π₯ = 0, π₯ = 1 revolved about the x-axis
180. π¦ = π₯1/3 β 2, π¦ = 0, π₯ = 8, π₯ = 27 revolved about π¦ = 4
181. π¦ = ββπ₯ β 2, π¦ = 0, π₯ = β6, π₯ = β2 revolved about π¦ = β1
182. π¦ = βπ₯ β 2, π¦ = β10 β π₯, π¦ = 0 revolved about π¦ = β6
ππ β« π(π)π ππ
π
π
ππ β π. πππ
ππ β« (π β π)(π + π)ππ ππ
π
πππ
ππ β πππ. ππ
ππ β« (π + π)(βππ β π)π ππ
π
ππ
ππ β ππ. πππ
ππ β« (π + π) ((ππ β ππ) β (ππ + π)) π ππ
π
ππππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~26~ NJCTL.org
Volume: Shell Method β Homework
Use the Shell Method to calculate the volume of the object created by rotating the described
region about the given axis.
183. π¦ = 1 β 2π₯ + 3π₯2 β 2π₯3, π¦ = 0, π₯ = 0, π₯ = 1 revolved about the y-axis
184. π¦ = 9 β π₯2, π¦ = 9 β 3π₯ revolved about the y-axis
185. π¦ = π₯β1/2, π¦ = 0, π₯ = 1, π₯ = 4 revolved about π₯ = β3
186. π¦ = sin(π₯2) , π¦ = 0, π₯ = 0, π₯ = βπ revolved about y-axis
ππ β« π(π β ππ + πππ β πππ)π ππ
π
ππ
πππ β π. πππ
ππ β« π ((π β ππ) β (π β ππ)) π ππ
π
ππ
ππ β ππ. πππ
ππ β« (π + π)(πβπ/π)π ππ
π
ππ
ππ β ππ. πππ
ππ β« π π¬π’π§(ππ ) π πβπ
π
ππ β π. πππ
Pre-Calc Intro to Integrals KEY ~27~ NJCTL.org
187. π¦ = 3π₯ β 1, π¦ = 2, π₯ = 1, π₯ = 3 revolved about the x-axis
188. π¦ = π₯3, π¦ = π₯ revolved about the x-axis
189. π¦ = βπ₯ β 2, π¦ = 0, π₯ = 1, π₯ = 4 revolved about π¦ = 5
190. π¦ = βπ₯ β 1, π¦ = 7 β π₯, π¦ = 0 revolved about π¦ = β10
ππ β« π (π
ππ +
π
π) π π
π
π
ππππ β πππ. ππ
ππ β« π(ππ/π β π)π ππ
π
π
πππ β π. πππ
ππ β« (βπ + π)(π + π)ππ ππ
βπ
ππ
ππ β ππ. πππ
ππ β« (π + ππ) ((π β π) β (ππ + π)) π ππ
π
πππ
ππ β πππ. πππ
Pre-Calc Intro to Integrals KEY ~28~ NJCTL.org
Unit Review Multiple Choice
1. β«4π₯+2
4π₯
4
1ππ₯ =
a. ln 4
b. 1
2ln 4
c. 3 +1
2ln 4
d. 5 +1
2ln 4
e. 2
2. β« (π₯2 β 4π₯ + 7)ππ₯2
β2
a. 0
b. 16
3
c. 8
3
d. 100
3
e. β100
3
3. The area under π¦ =1
π₯ from π₯ = 1 to π₯ = π4 is split into two equal area regions by π₯ = π.
Find π.
a. 2
b. 2.5
c. π2
d. ln 4
e. π
4. πΉ(π₯) = β« (π‘2 β 1)2π₯
π₯ππ‘, πΉβ²(π₯) =
a. 2π₯2 β π₯
b. 4π₯2 β π₯
c. 8π₯2 β π₯
d. 3π₯2 β 1
e. 7π₯2 β 1
5. β« π₯3ππ₯3
1 is approximated using right rectangular approximation method (RRAM), with 4
equal partitions. Find the approximate area and state whether it is under or over estimate.
a. 17.641 u2; under estimate
b. 17.641 u2; over estimate
c. 20 u2; over estimate
d. 20 u2; under estimate
e. 27 u2; over estimate
C
D
C
E
E
Pre-Calc Intro to Integrals KEY ~29~ NJCTL.org
6. Using the trapezoid rule and π = 8, approximate β« (π₯2 β 6)ππ₯4
2
a. 6.688
b. 3.839
c. 7.647
d. 6.667
e. 13.366
7. β« π(π₯)ππ₯6
3= 4, β« π(π₯) = β8ππ₯
10
6, πππ β« π(π₯)ππ₯ = β5
10
8, then which of the following
statements is true?
a. β« π₯π(π₯)6
3ππ₯ = 4π₯
b. β« 2π(π₯)10
3ππ₯ = β6
c. β« (π(π₯)10
8β 3)ππ₯ = β8
d. β« 5π(π₯)8
3ππ₯ = 5
e. β« π(π₯)10
3ππ₯ = β9
8. The area of the region bounded by the curves π¦ = 5 β π₯2, π¦ = π₯2 β 5, π₯ = 1, πππ π₯ = 2 is
a. 16
3
b. 29
3
c. 34
3
d. 92
3
e. 32
9. The volume of solid formed by the region bound by π¦ = π₯2, π¦ = 0, πππ π₯ = 1 revolved
about the x-axis is
a. π
b. π
2
c. π
3
d. π
4
e. π
5
10. The volume of the solid formed by the region bound by π¦ = π₯2, π¦ = β2π₯ + 3, and the x-axis
revolved around π¦ = β1 is
a. Ο β« (x2 β 1)21
0dx + Ο β« (β2x + 2)2dx
1.5
1
b. Ο β« ((x2)2 β 1)1
0dx + Ο β« ((β2x + 3)2 β 1)dx
1.5
1
c. Ο β« ((x2 + 1)2 β 1)1
0dx + Ο β« ((β2x + 4)2 β 1)dx
1.5
1
d. Ο β« ((x2 β 1)2 β 1)1
0dx + Ο β« ((β2x + 2)2 β 1)dx
1.5
1
e. Ο β« (x2)21
0dx + Ο β« (β2x + 2)2dx
1.5
1
A
D
A
E
C
Pre-Calc Intro to Integrals KEY ~30~ NJCTL.org
11. The area of the region bounded by the curves π¦2 = π₯ πππ π¦ = βπ₯ + 4 is
a. 11.682
b. 10.600
c. 6.486
d. 5.796
e. 5.408
12. The volume of solid formed by the region bound by π¦ = π₯2, π¦ = 0, πππ π₯ = 2 revolved
about the x=2 is
a. 25.133
b. 16.755
c. 8.378
d. 6.283
e. 5.924
13. The volume of the solid formed by the region bound by π¦ = π₯2 + 1, π¦ = βπ₯2 + 3, πππ π₯ = 0
revolved around π¦ = 4 is
a. 3.161
b. 5.194
c. 11.854
d. 13.433
e. 16.755
14. Use substitution to evaluate β«(π₯2 + 2)βπ₯3 + 6π₯ β 5ππ₯
a. 2
3(
1
4π₯4 + 3π₯2 β 5π₯)
3/2+ πΆ
b. 2
3(π₯3 + 6π₯ β 5)3/2 + πΆ
c. (π₯3 + 6π₯ β 5)3/2 + πΆ
d. 2
9(π₯3 + 6π₯ β 5)3/2 + πΆ
e. 2
9(3π₯2 + 6)3/2 + πΆ
15. Use substitution to evaluate β« π₯(2π₯2 β 7)5ππ₯3
1
a. 1,755,936
b. 73,164
c. 438,984
d. 292,656
e. 26,321
A
C
E
D
B
Pre-Calc Intro to Integrals KEY ~31~ NJCTL.org
Extended Response
1. Use the graph to answer the following:
a. β« π(π₯)ππ₯5
0
b. β« π(π₯)ππ₯β4
0
c. π(0)
d. πβ²(0)
2. The table represents the fuel consumption of a car at given times.
a. Approximate the fuel consumption using MRAM for 0 β€ π‘ β€ 8 and 4 rectangles.
b. What is the approximate rate of change in the fuel consumption at π‘ = 1?
c. If the maximum rate of fuel consumption occurs at π‘ = 5 min, what is the rate of
change in the fuel consumption at π‘ = 5? Explain.
time(min) 0 1 2 3 4 5 6 7 8
gal/min 2 3 4 2 3 5 3 4 2
π
βπ
π
π
28 gallons
1 gal/min per min
0; because it is a max
Pre-Calc Intro to Integrals KEY ~32~ NJCTL.org
3. The graph of a velocity function, πβ²(π₯), is shown
a. How far does the particle travel for 3 β€ π₯ β€ 7?
b. What is the particles acceleration at π₯ = 5?
c. Is particle speed increasing or decreasing at π₯ = 6? Explain.
4. Region R is bound by π¦ = π₯ + 3, π¦ = 9 β π₯2 and π₯ = 1.
a. find the area of R
b. Find the volume of the solid created by rotating R about π₯ = β1 using the
Washer Method
c. Find the volume of the solid created by rotating R about π₯ = 4 using the Shell
Method
ππ. ππ
π
Decreasing; the slope is negative
ππ
πβ π. πππ
ππ
πβ πππ. ππ
ππ
πβ ππ. πππ
Recommended