Berkeley City College
Homework Due:_____________
Calculus I - Math 3A - Chapter 5 - Integrals
Name___________________________________
Find the value of the indicated sum.
4
1)
∑
2 cos k = 1
π
k
1)
Objective: (5.1) Find Value of Sum
3
2)
∑
k = 1
3π
(-1)k sin 2
2)
Objective: (5.1) Find Value of Sum
Write the indicated sum in sigma notation.
6 6
6
3) 6 + + + . . . + 2 3
10
3)
Objective: (5.1) Write Sum in Sigma Notation
4) -(1 - 5)2 + (2 - 5)2 - (3 - 5)2
4)
Objective: (5.1) Write Sum in Sigma Notation
Find the value of the specified finite sum.
n
∑
5) Given a k = -6 and n
∑
b k = 5, find k = 1
k = 1
n
∑
(a k + b k ) .
5)
(a k - 2bk ) .
6)
k = 1
Objective: (5.1) Use Properties of Sigma Notation
n
∑
6) Given k = 1
a k = 2 and n
∑
k = 1
n
∑
b k = 7, find k = 1
Objective: (5.1) Use Properties of Sigma Notation
Instructor: K. Pernell
1
Use Special Sum Formulas to find the sum.
24
7)
∑
(8k 3 - 12k 2)
7)
k = 1
Objective: (5.1) Use Special Sum Formulas to Find Sum
Find the area under the curve of the function on the stated interval. Do so by dividing the interval into n equal
subintervals and finding the area of the corresponding circumscribed polygon.
8) f(x) = 2x2 + x + 3 from x = 0 to x = 6; n = 6
8)
Objective: (5.1) Find Area Under Curve Using Finite Subintervals
Find the formula and limit as requested.
9) For the function f(x) = 6x2 + 1, find a formula for the upper sum obtained by dividing the
interval [0, 3] into n equal subintervals. Then take the limit as n →∞ to calculate the area
under the curve over [0, 3].
9)
Objective: (5.1) Find Area Under Curve as n Approaches Infinity
n
Calculate the Riemann sum ∑
f(xi) △xi for the given data.
i = 1
10) f(x) = x + 4; P: 2 < 2.75 < 3.25 < 4.5 < 5.5; x1 = 2, x2 = 3, x3 = 4.25, x4 = 5
10)
Objective: (5.2) Find Riemann Sum from Given Data
Use the given values of a and b and express the given limit as a definite integral.
n
2
11) lim
∑ (3 c k - 11ck + 3) △xk , a = -5, b = 4
P →0 k = 1
11)
Objective: (5.2) Express Limit of Riemann Sum as a Definite Integral
12)
n
lim
∑
P →0 k = 1
2
c k + 8 △xk , a = -7, b = 2
12)
Objective: (5.2) Express Limit of Riemann Sum as a Definite Integral
2
Evaluate the definite integral using the definition.
5
13)
3 dx
-3
∫
13)
Objective: (5.2) Evaluate Definite Integral Using Definition
14)
∫
4
(-2x + 8) dx
14)
-8
Objective: (5.2) Evaluate Definite Integral Using Definition
Solve the problem.
15) The velocity of an object is given by the velocity function v(t) = t/42. If the object is at the
origin at time t = 0, find the position at time t = 20.
15)
Objective: (5.2) Find Position Given Velocity Function
16) An objectʹs velocity function is graphed below. Determine the objectʹs position at time
t = 40 assuming the object is at the origin at time t = 0.
16)
v
6
5
4
3
2
1
20
40
60
80
100
120
t
Objective: (5.2) Find Position Given Graph of Velocity Function
17) Suppose that f and g are continuous and that ∫
11
f(x) dx = -4 and Find ∫
∫
7
7
11
4f(x) + g(x) dx .
7
Objective: (5.3) Evaluate Definite Integral Using Properties
3
11
g(x) dx = 9.
17)
Find Gʹ(x).
18) G(x) = ∫
x4
18)
sin t dt
0
Objective: (5.3) Differentiate Integral
19) G(x) = x
∫
18t7 dt
19)
1
Objective: (5.3) Differentiate Integral
20) G(x) = ∫
x
4x + 3 dt
20)
0
Objective: (5.3) Differentiate Integral
21) G(x) = ∫
x4
cos t dt
21)
0
Objective: (5.3) Differentiate Integral
22) f(x) = ∫
0
x
1
du
u2
22)
Objective: (5.3) Find Intervals of Monotonicity/Concavity
4
Answer the question.
23) Consider the function G(x) = ∫
x
f(t) dt , where f(t) oscillates about the line y = 2 over the
23)
0
x-region [0, 12].
The graph is given below:
y
15
10
5
1
2
3
4
5
6
7
8
9
10 11 12 t
-5
-10
-15
At what values of x over this region do the local maxima of G(x) occur?
Objective: (5.3) Analyze Graph of Function
Evaluate the integral.
1
8 2
24)
t dt
∫
24)
0
Objective: (5.4) Evaluate Definite Integral
25)
∫
4
2 x dx
25)
0
Objective: (5.4) Evaluate Definite Integral
26)
∫
3π/2
14 cos x dx
26)
π/2
Objective: (5.4) Evaluate Definite Integral
5
27)
∫ x7(x8 - 10)4 dx
27)
Objective: (5.4) Find Integral Using Substitution
28)
∫ 10x2
4
3 + 4x3 dx
28)
Objective: (5.4) Find Integral Using Substitution
29)
∫ x2 cos (x3 + 6) sin (x3 + 6) dx
29)
Objective: (5.4) Find Integral Using Substitution
Evaluate the definite integral.
2
30)
(x + 2)3 dx
0
∫
30)
Objective: (5.4) Evaluate Definite Integral Using Substitution
31)
∫
6
5
(6 + x)2
0
dx
31)
Objective: (5.4) Evaluate Definite Integral Using Substitution
32)
∫
1
6 x dx
32)
16 + 3x2
0
Objective: (5.4) Evaluate Definite Integral Using Substitution
33)
∫
2π
3 cos2 x sin x dx
33)
π/3
Objective: (5.4) Evaluate Definite Integral Using Substitution
34)
∫
π
(1 + cos 7t) 2 sin 7t dt
34)
0
Objective: (5.4) Evaluate Definite Integral Using Substitution
6
Solve the problem.
35) A certain company has found that its expenditure rate per day (in hundreds of dollars) on
a certain type of job is given by Eʹ(x) = 6x + 9, where x is the number of days since the start
of the job. Find the expenditure if the job takes 8 days.
35)
Objective: (5.4) Solve Apps: Fundamental Theorem of Calculus
36) A certain object moves in such a way that its velocity (in m/s) after time t (in s) is given by
V(t) = t2 + 2t + 8. Find the distance traveled during the first four seconds.
36)
Objective: (5.4) Solve Apps: Fundamental Theorem of Calculus
Use symmetry to help evaluate the integral.
π/3
(cos x + sin x) dx
37)
-π/3
∫
37)
Objective: (5.5) Evaluate Integral Using Symmetry
38)
∫
1
(2 + 4x + 3x2 + x3) dx
38)
-1
Objective: (5.5) Evaluate Integral Using Symmetry
Use the requested method to approximate the definite integral.
39) Use the method of left Riemann sum with n = 4 to approximate the value of ∫
7
x2 dx
39)
3
Objective: (5.6) Use Riemann Sum to Approximate Integral
40) Use the method of right Riemann sum with n = 4 to approximate the value of ∫
5
x2 dx
40)
1
Objective: (5.6) Use Riemann Sum to Approximate Integral
41) Use the method of midpoint Riemann sum with n = 4 to approximate the value of
5
x2 dx
1
∫
Objective: (5.6) Use Riemann Sum to Approximate Integral
7
41)
Express the limit as a definite integral where P is a partition of the given interval.
n
2
42) lim P →0 ∑ (3 c k - 9ck + 9) △xk , [-10, 4]
k = 1
42)
Objective: (5.3) Express Limit of Riemann Sums as Definite Integral
n
43) lim P →0
∑
5
c k △xk , [-5, 4]
43)
k = 1
Objective: (5.3) Express Limit of Riemann Sums as Definite Integral
8
Answer Key
Testname: 13FALL_CH5_PROBS
π
π
π
1) 2 cos π + 2 cos + 2 cos + 2 cos = -1 + 2
2
3
4
2) -sin 3π
3π
3π
+ sin - sin = 1
2
2
2
10
∑
3) 6
i = 1
3
∑
5)
6)
7)
8)
k = 1
-1
-12
661,200
221
32) 2 19 - 8
7
33) - 8
(-1)k (k - 5)2
34)
324n 3 + 486n 2 + 162n
; Area = 57
6n 3
∫
∫
∫
∫
16) 160
17) -7
18) 4x3 sin (x4)
19) 9x3
20) 4x + 3
21) 4x3 cos (x2)
22) f(x) is increasing on (-∞, 0) and (0, ∞)
f(x) is concave up on (-∞, 0)
23) ≈ 2.1, ≈8.0
1
24)
1536
32
3
26) -28
(x8 - 10) 5
27)
+ C
40
28)
8
21
35) $26,400
36) 69.3 m
37) 3
38) 6
39) 86
40) 54
41) 41
4
42)
(3x2 - 9x + 9) dx
-10
4
x5 dx
43)
-5
10) 27.3125
4
11)
(3x2 - 11x + 3) dx
-5
2
x2 + 8 dx
12)
-7
13) 24
14) 144
100
15)
21
25)
2
(sin (x3 + 6))3/2 + C
9
30) 60
5
31)
12
1
i
4)
9) 3 + 29)
2
(3 + 4x3) 5/4 + C
3
9