Calculus2
The Fundamental Theorem of Calculus (Part 2)
1. State the Fundamental Theorem of Calculus (Part 2), including conditions.
� 2–7 Use the FTC (Part 2) to compute the definite integral.
2.
∫ 3
−1
x5 dx 3.
∫ 2
0
(6x2 − 4x + 5) dx
4.
∫ 2π
π
cos θ dθ 5.
∫ π/4
0
sec2θ dθ
6.
∫ e
1
1
2t dt 7.
∫ 9
1
t + 1 √
t dt
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. This problem explores the following question:
∫ b
a
f(x)g(x) dx ? = F(x)G(x)
∣
∣
∣
∣
b
a
(1)
To this end, consider the integral
∫ 2
0
(x + 1)(x − 3) dx
a) Compute the integral using Equation 1.
b) Compute the integral by first expanding the integrand.
c) State whether Equation 1 is True or False.
9. This problem explores the following question:
∫ b
a
[f(x)]n dx ? =
[f(x)]n+1
n + 1
∣
∣
∣
∣
b
a
(2)
To this end, consider the integral
∫ 1
0
(3x + 1)2 dx
a) Compute the integral using Equation 2.
b) Compute the integral by first expanding the integrand.
c) State whether Equation 2 is True or False.
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10. The area of region B is three times the area of region A. Express b in terms of a.
x
y
A
y = ex
a x
y
B
y = ex
b
Applications in Net Change
� 11, 12 In one sentence, interpret the definite integral in the given scenario.
11. Oil leaks from a tank at a rate of r(t) gallons per minute at time t.
∫ 120
0
r(t) dt
12. The current I in a wire at a point x is defined as the rate at which the charge Q is changing at x: I(x) = Q′(x).
∫ b
a
I(x) dx
13. At a distance of x miles from the start of a trail, the elevation increases at a rate of 900x feet per mile. Find the net change in elevation over the first three miles.
14. A particle moves along a line with velocity v(t) = t2 − 2t − 8 meters per second.
a) Denoting the particle’s position by s(t), find the net displacement of the particle over [1, 6].
b) Find the total distance traveled by the particle over the same time interval.
Hint on Part (b): If the particle turns around one or more times, you’ll need to compute the distance traveled in each direction, and then sum.
History
15. Citing your reference source (e.g. Wikipedia), state one interesting fact about Isaac Newton.
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Solutions to Selected Problems
1. FTC (Part 2)
2. 364
3
3. 18
4. 0
5. 1
6. 1
2
7. 64
3
8. a) -16
b) −22
3
c) False
9. a) 21
b) 7
c) False
10. b = ln(3ea − 2)
11. The net change in oil over two hours.
12. The net change in charge over the segment of wire from x = a to x = b.
13. 4050 ft
14. a) −10
3 m
b) 98
3 m
15. Newton fact.
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