Section 5.1 Anti-derivatives / Integration
#1-32: Find the following antidervatives, be sure to include the plus “C” in your answer.
1) ∫ 3𝑥 2 𝑑𝑥
1
2) ∫ 2𝑥 3 𝑑𝑥
1
3) ∫ 3 𝑥𝑑𝑥
4) ∫ 2 𝑥𝑑𝑥
5) ∫ 2𝑑𝑥
6) ∫ 7𝑑𝑥
7) ∫(6𝑥 + 5)𝑑𝑥
8) ∫(4𝑥 3 − 3𝑥 2 + 5)𝑑𝑥
3
9) ∫ √𝑥 2 𝑑𝑥
10) ∫ √𝑥 5 𝑑𝑥
11) ∫ 𝑦 2⁄3 𝑑𝑦
12) ∫ 𝑦 3⁄5 𝑑𝑦
13) ∫(𝑦 3⁄2 + 𝑦 1⁄2 )𝑑𝑦
14) ∫(𝑦 2⁄3 − 𝑦 1⁄3 )𝑑𝑦
5
15) ∫ 2 𝑑𝑥
𝑥
2
3
16) ∫ 4 𝑑𝑥
𝑥
3
17) ∫ 3𝑥 2 𝑑𝑥
18) ∫ 4𝑥 3 𝑑𝑥
19) ∫(4𝑧 −3 + 3𝑧 −2 )𝑑𝑧
20) ∫(7𝑧 −3 + 5𝑧 −2 )𝑑𝑧
21) ∫ 2𝑦(𝑦 2 + 3)𝑑𝑦
22) ∫ 4𝑦(𝑦 3 + 3𝑦 − 4)𝑑𝑦
23) ∫(3𝑥 − 4)2 𝑑𝑥
24) ∫(𝑥 2 + 5)2 𝑑𝑥
25) ∫(𝑧 + 1) (𝑧 − 4)𝑑𝑧
26) ∫(𝑦 − 4) (𝑦 + 7)𝑑𝑦
27) ∫
3𝑥 2 +2𝑥
𝑑𝑥
𝑥
28) ∫
4𝑥 2 −5𝑥−3
𝑑𝑥
𝑥
29) ∫
𝑥 3⁄2 +4𝑥 1⁄2
𝑑𝑥
𝑥 1⁄2
30) ∫
2𝑥 4⁄3 +3𝑥 1⁄3
𝑑𝑥
𝑥 4⁄3
3
31) ∫ (𝑥 2 + 5𝑥 2 ) 𝑑𝑥
4
32) ∫ (𝑥 5 + 3) 𝑑𝑥
#33 – 34 Evaluate the special anti-derivatives / integrals
33) ∫ 𝑒 𝑥 𝑑𝑥
1
34) ∫ 𝑥 𝑑𝑥
Section 5.2 Integration by substitution
#1 – 18: Use u-substitution to evaluate the indefinite integrals.
1) ∫ 2𝑥(𝑥 2 + 5)4 𝑑𝑥
2) ∫ 3𝑥 2 (𝑥 3 − 4)5 𝑑𝑥
3) ∫(2𝑥 + 3) (𝑥 2 + 3𝑥 − 4)3 𝑑𝑥
4) ∫(8𝑥 + 2) (4𝑥 2 + 2𝑥 − 7)5 𝑑𝑥
2𝑥
5) ∫ 2 𝑑𝑥
√𝑥 +5
6) ∫ 3
3𝑥 2
√𝑥 3 +4
3𝑥 2 +6𝑥+1
7) ∫ 3 2 𝑑𝑥
√𝑥 +3𝑥 +𝑥
8𝑥 3 +6𝑥+5
8) ∫ 3
√2𝑥 4 +3𝑥 2 +5𝑥+1
2
𝑑𝑥
3
9) ∫ 2𝑥𝑒 𝑥 𝑑𝑥
10) ∫ 3𝑥 2 𝑒 𝑥 𝑑𝑥
11) ∫(2𝑥 + 5) 𝑒 𝑥
2 +5𝑥
𝑑𝑥
1
12) ∫(6𝑥 2 + 8𝑥) 𝑒 2𝑥
3 +4𝑥 2 +3
𝑑𝑥
2
13) ∫ (𝑥−4)3 𝑑𝑥
3
𝑑𝑥
14) ∫ (2𝑥+5)3 𝑑𝑥
3
15) ∫(3𝑥) ∗ √ 𝑥 2 + 4 𝑑𝑥
2
5
16) ∫ 5𝑥 ∗ √ 𝑥 2 − 3 𝑑𝑥
2
1
1
17) ∫(𝑥 2 + 6𝑥 − 3)√3 𝑥 3 + 3𝑥 2 − 3𝑥 + 1𝑑𝑥
18) ∫(𝑥 3 + 10𝑥 − 4)√4 𝑥 4 + 5𝑥 2 − 4𝑥 + 3𝑑𝑥
#19-36: Use u-substitution to evaluate the indefinite integrals.
19) ∫ 6𝑥(𝑥 2 + 5)4 𝑑𝑥
20) ∫ 15𝑥 2 (𝑥 3 − 4)5 𝑑𝑥
21) ∫(4𝑥 + 6) (𝑥 2 + 3𝑥 − 4)3 𝑑𝑥
22) ∫(32𝑥 + 8) (4𝑥 2 + 2𝑥 − 7)5 𝑑𝑥
24𝑥 2
10𝑥
23) ∫ 2 𝑑𝑥
√𝑥 +5
24) ∫ 3
√𝑥 3 +4
9𝑥 2 +18𝑥+3
25) ∫ 3 2 𝑑𝑥
√𝑥 +3𝑥 +𝑥
2
26) ∫ 3
3
2
𝑑𝑥
3
28) ∫ 2 𝑥 2 𝑒 𝑥 𝑑𝑥
29) ∫(16𝑥 + 40) 𝑒 𝑥
2 +5𝑥
𝑑𝑥
9
30) ∫(48𝑥 2 + 64𝑥) 𝑒 2𝑥
3 +4𝑥 2 +3
14
31) ∫ (𝑥−4)3 𝑑𝑥
3
16𝑥 3 +12𝑥+10
√2𝑥 4 +3𝑥 2 +5𝑥+1
27) ∫ 3 𝑥𝑒 𝑥 𝑑𝑥
3
𝑑𝑥
32) ∫ (2𝑥+5)3 𝑑𝑥
3
33) ∫ (5 𝑥) ∗ √2 𝑥 2 + 4 𝑑𝑥
5
1
35) ∫(3𝑥 2 + 18𝑥 − 9)√3 𝑥 3 + 3𝑥 2 − 3𝑥 + 1𝑑𝑥
1
5
34) ∫ 4 𝑥 ∗ √2 𝑥 2 − 3 𝑑𝑥
36) ∫(2𝑥 3 + 20𝑥 − 8)√ 𝑥 4 + 5𝑥 2 − 4𝑥 + 3𝑑𝑥
4
𝑑𝑥
Section 5.2 Integration by substitution
#37- 50: Use u-substitution to evaluate the indefinite integrals that involve the ln(x).
𝑎
Hint ∫ 𝑑𝑥 = 𝑎𝑙𝑛|𝑥| + 𝐶 and ∫ 𝑎𝑒 𝑥 𝑑𝑥 = 𝑎𝑒 𝑥 + 𝐶
𝑥
2𝑥
37) ∫ 𝑥 2 +3 𝑑𝑥
3𝑥 2 +8𝑥
39) ∫ 𝑥 3 +4𝑥2 +1 𝑑𝑥
−3𝑥 −4
41) ∫ −3 𝑑𝑥
𝑥 +5
3𝑥 2
38) ∫ 𝑥 3 +5 𝑑𝑥
20𝑥 3 +6𝑥+2
40) ∫ 5𝑥 4 +3𝑥2 +2𝑥 𝑑𝑥
−6𝑥 −4 + 5
42) ∫ −3
𝑑𝑥
2𝑥 +5𝑥−4
43 and 44 have been deleted
8𝑥
45) ∫ 2 𝑑𝑥
𝑥 +3
6𝑥 2 +16𝑥
47) ∫ 3 2 𝑑𝑥
𝑥 +4𝑥 +1
−6𝑥 −4
49) ∫ −3 𝑑𝑥
𝑥 +5
12𝑥 2
46) ∫ 3 𝑑𝑥
𝑥 +5
10𝑥 3 +3𝑥+1
48) ∫ 4 2
𝑑𝑥
5𝑥 +3𝑥 +2𝑥
−18𝑥 −4 + 15
50) ∫ −3
𝑑𝑥
2𝑥 +5𝑥−4
Section 5.3 Area and the definite integral.
#1-10: Use the trapezoid rule to estimate the area between the curve and the x-axis.
Specifically:
a) Use your calculator to create a graph of the function.
𝑥 −𝑥
b) Subdivide the x-axis into n-intervals. (Each interval will have a width; 2 𝑛 1)
c) Evaluate the function at each value marked on the x-axis in step b.
d) Draw the trapezoids implied by the points plotted.
e) Find the area of each trapezoid.
f) Add the individual areas to get an estimate of the area under the curve.
𝑥
g) Use your calculator to calculate the exact area under the curve ∫𝑥 2 𝑓(𝑥)𝑑𝑥
1
(Round the y-coordinates to 2-decimal places when appropriate.)
1) 𝑓(𝑥) =
60
;
𝑥
using 2 intervals from x1 = 2 to x2 = 10
2) 𝑓(𝑥) =
40
;
𝑥
using 2 intervals from x1 = 2 to x2 = 8
3) 𝑓(𝑥) =
20
;
𝑥
using 4 intervals from x1 = 2 to x2 = 10
4) 𝑓(𝑥) =
40
;
𝑥
using 4 intervals from x1 = 5 to x2 = 13
5) f(x) = x2 + 2; using n = 2 intervals from x1 = 1 to x2 = 3.
6) f(x) = x2 + 1; using n = 2 intervals from x1 = 0 to x2 = 2.
7) f(x) = x2 + 2; using n = 4 intervals from x1 = 1 to x2 = 3.
8) f(x) = x2 + 1; using n = 4 intervals from x1 = 0 to x2 = 2.
9) f(x) = 12 – x2; using n = 3 intervals from x1 = 0 to x2 = 3
10) f(x) = 10 – x2; using n = 3 intervals from x1 = 0 to x2 = 3
Section 5.4 The Fundamental Theorem of Calculus
#1-30:
a) Use your calculator to evaluate the definite integral.
b) Use the Fundamental Theorem of Calculus to evaluate the definite integral.
5
2) ∫1 (2𝑥 − 5)𝑑𝑥
3
4) ∫0 (8𝑥 3 − 6𝑥 2 + 2𝑥)𝑑𝑥
1) ∫2 (4𝑥 − 3)𝑑𝑥
3) ∫0 (4𝑥 3 + 3𝑥 2 − 7)𝑑𝑥
9
6
2
16
5) ∫1 √𝑥𝑑𝑥
6) ∫1 3√𝑥𝑑𝑥
25
7) ∫4 (√𝑥 − 4𝑥)𝑑𝑥
9
8) ∫1 (2√𝑥 − 6𝑥)𝑑𝑥
𝑒2 1
2
9) ∫0 𝑒 𝑥 𝑑𝑥
10) ∫𝑒
𝑥
𝑑𝑥
4
12) ∫−2(2𝑥 − 4)(𝑥 2 − 4𝑥 + 1)5 𝑑𝑥
4
14) ∫−2(6𝑥 − 12)(𝑥 2 − 4𝑥 + 1)2 𝑑𝑥
11) ∫−3(2𝑥) (𝑥 2 − 5)3 𝑑𝑥
13) ∫−3(6𝑥) (𝑥 2 − 5)3 𝑑𝑥
𝑒 3 2𝑥
𝑑𝑥
𝑥 2 +3
2 3𝑥 2 +3
2
19) ∫0
2
𝑥 3 +3𝑥
16) ∫𝑒 2
2 4𝑥 3 +5
𝑑𝑥
1
𝑑𝑥
(𝑥−4)3
𝑥
8
27) ∫1 𝑥 2⁄3 𝑑𝑥
7
29) ∫−4 5 𝑑𝑥
0
4
𝑑𝑥
22) ∫3
3
5 3𝑥 2 +2𝑥−4
𝑥 4 +5𝑥
4 12𝑥 2
𝑑𝑥
𝑥 3 +5
2𝑥
23) ∫0 3𝑥 2 𝑒 𝑥 𝑑𝑥
25) ∫3
18) ∫1
20) ∫−2 (4𝑥−12)3 𝑑𝑥
21) ∫−2 2 𝑑𝑥
𝑥 +3
1
1
𝑒 5 3𝑥 2
𝑑𝑥
𝑥 3 +5
15) ∫𝑒
17) ∫1
1
𝑑𝑥
1
2
24) ∫0 2𝑥𝑒 𝑥 𝑑𝑥
6 4𝑥 2 −5𝑥−3
26) ∫2
𝑥
16
28) ∫1 𝑥 3⁄4 𝑑𝑥
3
30) ∫−2 4𝑑𝑥
𝑑𝑥
Section 5.5 The Fundamental Theorem of Calculus part 2
You may use your calculator to find the areas that are needed in problems 1- 4.
1) Find the shaded area.
0
1a) ∫−1(𝑥 2 − 4𝑥)𝑑𝑥
4
1b) |∫0 (𝑥 2 − 4𝑥)𝑑𝑥 |
0
4
1c) ∫−1(𝑥 2 − 4𝑥)𝑑𝑥 + |∫0 (𝑥 2 − 4𝑥)𝑑𝑥 |
Section 5.5 The Fundamental Theorem of Calculus part 2
2) Find the shaded area.
0
2a) ∫−1(𝑥 2 − 6𝑥)𝑑𝑥
6
2b) |∫0 (𝑥 2 − 6𝑥)𝑑𝑥 |
0
6
2c) ∫−1(𝑥 2 − 6𝑥)𝑑𝑥 + |∫0 (𝑥 2 − 6𝑥)𝑑𝑥 |
Section 5.5 The Fundamental Theorem of Calculus part 2
3) Find the shaded area.
2
3a) ∫1 (𝑒 𝑥 − 𝑒)𝑑𝑥
1
3b) |∫−2(𝑒 𝑥 − 𝑒)𝑑𝑥 |
2
1
3c) ∫1 (𝑒 𝑥 − 𝑒)𝑑𝑥 + |∫−2(𝑒 𝑥 − 𝑒)𝑑𝑥 |
Section 5.5 The Fundamental Theorem of Calculus part 2
4) Find the shaded area.
3
4a) ∫0 (1 − 𝑒 −𝑥 )𝑑𝑥
0
4b) |∫−2(1 − 𝑒 −𝑥 )𝑑𝑥 |
3
0
4c) ∫0 (1 − 𝑒 −𝑥 )𝑑𝑥 + |∫−2(1 − 𝑒 −𝑥 )𝑑𝑥|
Section 5.5 The Fundamental Theorem of Calculus part 2
#5 – 17:
a) Sketch a graph of the function f(x) over the given interval [a,b].
b) Find any x-intercept within the interval [a,b].
c) Find the area between the x-axis and f(x) over the interval [a,b] using definite integrals.
5) f(x) = 3x – 15; [4,8]
6) f(x) = 2x – 12; [4,9]
7) f(x) = 2 – 2x2; [0,3]
8) f(x) = 4 – 4x2; [0,4]
9) f(x) = x3; [-2,1]
10) f(x) = x5; [-1,1]
11) 𝑓(𝑥) = 𝑥 2 − 4; [-1,3]
12) 𝑓(𝑥) = 𝑥 2 − 9; [-1,5]
You may use your calculator to find the integrals needed for problems 11 and 12.
13 and 14 have been deleted.
Decimal answers are okay for problems 15 and 16.
15) 𝑓(𝑥) = 3√𝑥 − 𝑥; [7,12]
16) 𝑓(𝑥) = 2√𝑥 − 𝑥; [2,7]
17) 𝑓(𝑥) = 3𝑥 − 15; [5,8]
18) f(x) = 5x – 20; (3,6)
Section 5.6: The Area Between Two Curves
Use your calculator to find the requested area in problems #1-10.
1) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = 4
The function whose graph is represented by the solid line is g(x) = x2
2) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = 6
The function whose graph is represented by the solid line is g(x) = x2 + 2
Section 5.6: The Area Between Two Curves
3) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = -x2 + 8
The function whose graph is represented by the solid line is g(x) = x2
4) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = -x2 + 18
The function whose graph is represented by the solid line is g(x) = x2
Section 5.6: The Area Between Two Curves
5) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = -x2 + 11
The function whose graph is represented by the solid line is g(x) = -x+4
6) Find the shaded area.
The function whose graph is represented by the dashed is f(x) = -x2 + 7
The function whose graph is represented by the solid line is g(x) = -x+3
Section 5.6: The Area Between Two Curves
7) Find the shaded area.
The function whose graph is represented by the dashed is 𝑓(𝑥) = √𝑥
The function whose graph is represented by the solid line is 𝑔(𝑥) = 𝑥 2
8) Find the shaded area.
The function whose graph is represented by the dashed is 𝑓(𝑥) = √𝑥
The function whose graph is represented by the solid line is 𝑔(𝑥) = 𝑥 3
Section 5.6: The Area Between Two Curves
9) Find the shaded area.
The function whose graph is represented by the dashed is 𝑓(𝑥) = 2 − 𝑥
The function whose graph is represented by the solid line is 𝑔(𝑥) = 𝑥 3 + 3𝑥 2
(Use the intersect feature on your calculator to estimate the points of intersection when needed. Round
to 2 decimals.)
10) Find the shaded area.
The function whose graph is represented by the dashed is 𝑓(𝑥) = 4 − 2𝑥
The function whose graph is represented by the solid line is 𝑔(𝑥) = 𝑥 3 − 18𝑥 2 + 17𝑥
(Use the intersect feature on your calculator to estimate the points of intersection when needed. Round
to 2 decimals.)
Section 5.6 The Area Between Two Curves
#11-22: Determine the area of the region bounded by the given conditions.
11) 𝑓(𝑥) = 𝑥 − 5 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 + 𝑥 + 1 𝑜𝑛 [−2,1].
1
12) 𝑓(𝑥) = 2 𝑥 − 1 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 + 2𝑥 − 1 𝑜𝑛 [1,5].
13) 𝑓(𝑥) = 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 3√𝑥 𝑜𝑛 [0,1].
14) 𝑓(𝑥) = 𝑥 3 𝑎𝑛𝑑 𝑔(𝑥) = 4√𝑥 𝑜𝑛 [0,1].
15) 𝑓(𝑥) = 𝑥 2 − 4𝑥 𝑎𝑛𝑑 𝑔(𝑥) = −3𝑥 2 𝑜𝑛 [0,1].
16) 𝑓(𝑥) = 𝑥 2 − 9𝑥 𝑎𝑛𝑑 𝑔(𝑥) = −8𝑥 2 𝑜𝑛 [0,1].
17) 𝑓(𝑥) = 𝑥 + 1 𝑎𝑛𝑑 𝑔(𝑥) = 7 − 𝑥 𝑜𝑛 [2,5].
18) 𝑓(𝑥) = 𝑥 − 3 𝑎𝑛𝑑 𝑔(𝑥) = −𝑥 + 7 𝑜𝑛 [1,6].
19) 𝑓(𝑥) = 4𝑥 + 16 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥 2 + 10 𝑜𝑛 [−1,4].
20) 𝑓(𝑥) = 2𝑥 + 9 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 + 1 𝑜𝑛 [−2,5].
21) 𝑓(𝑥) = 2𝑥 2 + 6 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 + 5𝑥 𝑜𝑛 [2,6].
22) 𝑓(𝑥) = 𝑥 2 + 3𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥 2 − 2𝑥 + 4 𝑜𝑛 [1,6].
#23-30: Find the two points where the graphs of the functions intersect. Then find the area of the curve
that lies between the intersection points.
23) 𝑓(𝑥) = 9 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 + 6𝑥 − 7
24) 𝑓(𝑥) = 2 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 − 3𝑥 − 2
25) 𝑓(𝑥) = 9 − 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 5
26) 𝑓(𝑥) = 25 − 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 9
27) 𝑓(𝑥) = 2𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 6𝑥
28) 𝑓(𝑥) = 5𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 10𝑥
29) f(x) = 0 and g(x) = x2 – 4
30) f(x) = 0 and g(x) = x2 - 9
Chapter 5 Review
#1-3: Find the following antidervatives, be sure to include the plus “C” in your answer.
2
4
1) ∫ √𝑥 3 𝑑𝑥
2) ∫ 5𝑥3 𝑑𝑥
3) ∫
6𝑥 2 +2𝑥−4
𝑑𝑥
2𝑥
#4 – 11: Use u-substitution to evaluate the indefinite integrals.
3
4) ∫(6𝑥 + 4) (3𝑥 2 + 4𝑥 − 2)5 𝑑𝑥
5) ∫ 6𝑥 2 𝑒 2𝑥 𝑑𝑥
5
7) ∫ (𝑥−2)3 𝑑𝑥
8) ∫ 3
6𝑥 2
√𝑥 3 +4
9𝑥 2 +24𝑥
𝑑𝑥
6) ∫ 6𝑥(3𝑥 2 + 5)4 𝑑𝑥
6𝑥 2
9) ∫ 2𝑥3 +5 𝑑𝑥
11) ∫ 12𝑥 2 (𝑥 3 − 2)5 𝑑𝑥
10) ∫ 𝑥 3 +4𝑥2 +1 𝑑𝑥
12) f(x) = x2 + 3; using n = 4 intervals from x1 = 1 to x2 = 5.
Use the trapezoid rule to estimate the area between the curve and the x-axis.
Specifically:
a) Use your calculator to create a graph of the function.
𝑥 −𝑥
b) Subdivide the x-axis into n-intervals. (Each interval will have a width; 2 𝑛 1)
c) Evaluate the function at each value marked on the x-axis in step b.
d) Draw the trapezoids implied by the points plotted.
e) Find the area of each trapezoid.
f) Add the individual areas to get an estimate of the area under the curve.
𝑥
g) Use your calculator to calculate the exact area under the curve ∫𝑥 2 𝑓(𝑥)𝑑𝑥
1
#13-16:
a) Use your calculator to evaluate the definite integral. (round to 2 decimals)
b) Use the Fundamental Theorem of Calculus to evaluate the definite integral.
9
13) ∫1 4√𝑥𝑑𝑥
0
12
15) ∫−2 (4𝑥−5)3 𝑑𝑥
1
14) ∫−2(3𝑥 2 − 5)(𝑥 3 − 5𝑥 + 1)5 𝑑𝑥
1
2
16) ∫0 8𝑥𝑒 𝑥 𝑑𝑥
17) Find the shaded area. It is okay to have decimal answers and to use your calculator on problem 17.
1
2
The function whose graph is represented by the dashed is 𝑓(𝑥) = 𝑥
The function whose graph is represented by the solid line is 𝑔(𝑥) = √𝑥
18) 𝑓(𝑥) = 3 − 3𝑥 2 ; [-1,2]
a) Sketch a graph of the function f(x) over the given interval [a,b].
b) Find any x-intercept within the interval [a,b].
c) Find the area between the x-axis and f(x) over the interval [a,b] using definite integrals.
#19-20: Determine the area of the region bounded by the given conditions.
19) 𝑓(𝑥) = 𝑥 2 − 4𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 5 𝑜𝑛 [0,5].
20) 𝑓(𝑥) = 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = √𝑥 𝑜𝑛 [0,1].
21) f(x) = x2 + 3x and g(x) = 5 - x Find the two points where the graphs of the functions intersect. Then
find the area of the curve that lies between the intersection points.
Chapter 5 Practice Test Part 1
#1-3: Find the following antidervatives, be sure to include the plus “C” in your answer.
3
1) ∫ √𝑥 2 𝑑𝑥
1
2) ∫ 2𝑥4 𝑑𝑥
3) ∫
6𝑥 3 +2𝑥 2
𝑑𝑥
4𝑥 2
#4 – 9: Use u-substitution to evaluate the indefinite integrals.
4) ∫(6𝑥 + 5) (3𝑥 2 + 5𝑥)4 𝑑𝑥
24𝑥
6) ∫ (4𝑥2 −5)3 𝑑𝑥
4𝑥
8) ∫ 2𝑥2 +5 𝑑𝑥
2
5) ∫ 2𝑥𝑒 𝑥 𝑑𝑥
7
7) ∫ 𝑥−1 𝑑𝑥
√
12
9) ∫ 3𝑥+7 𝑑𝑥
10) f(x) = x2; using n = 2 intervals from x1 = 0 to x2 = 4.
Use the trapezoid rule to estimate the area between the curve and the x-axis.
Specifically:
a) Use your calculator to create a graph of the function.
𝑥 −𝑥
b) Subdivide the x-axis into n-intervals. (Each interval will have a width; 2 𝑛 1)
c) Evaluate the function at each value marked on the x-axis in step b.
d) Draw the trapezoids implied by the points plotted.
e) Find the area of each trapezoid.
f) Add the individual areas to get an estimate of the area under the curve.
𝑥
g) Use your calculator to calculate the exact area under the curve ∫𝑥 2 𝑓(𝑥)𝑑𝑥 (round to 2 decimals)
1
Chapter 5 Practice Test Part 2
#11-14:
a) Use your calculator to evaluate the definite integral. (round to 2 decimals)
b) Use the Fundamental Theorem of Calculus to evaluate the definite integral.
5
11) ∫1 (2𝑥)𝑑𝑥
0
3
13) ∫−2 (𝑥−5)2 𝑑𝑥
1
12) ∫0 (2𝑥)(𝑥 2 + 1)2 𝑑𝑥
3
14) ∫1 6𝑒 2𝑥 𝑑𝑥
15) Find the shaded area. It is okay to have decimal answers and to use your calculator on problem 15.
The function whose graph is represented by the dashed is 𝑓(𝑥) = 𝑥 + 1
The function whose graph is represented by the solid line is 𝑔(𝑥) = 7 − 𝑥
16) 𝑓(𝑥) = 𝑥 2 − 4; [-1,3]
a) Sketch a graph of the function f(x) over the given interval [a,b].
b) Find any x-intercept within the interval [a,b].
c) Find the area between the x-axis and f(x) over the interval [a,b] using definite integrals.
#17-18: Determine the area of the region bounded by the given conditions.
17) 𝑓(𝑥) = 𝑥 2 − 𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 7 𝑜𝑛 [1,3].
18) 𝑓(𝑥) = 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 𝑜𝑛 [3,5].
19) f(x) = x2 and g(x) = x Find the two points where the graphs of the functions intersect. Then find the
area of the curve that lies between the intersection points.