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Calculus
Integration (4.1-4.3) Quiz Review
What is the general power rule for integration?
List all the trigonometric antiderivatives (try from memory).
ò cos xdx = sinx + C
ò sin xdx = -cosx + C
ò sec x tan xdx
= secx
ò csc
2
ò sec xdx = tanx + C
ò csc x cot xdx = -cscx + C
2
xdx = -cot2x + C
Find the indefinite integral.
1.
ò (2x
2
+ x -1)dx
2x3/3 + x2/2 – x + C
2.
ò
x 3 +1
dx
x2
= (x + x-2) dx
3.
ò (5cos x - 2sec
2
5sinx – 2tanx + C
= x2/2 – x-1 + C
4. A ball is launched vertically upward from ground level with an initial velocity of 96 feet per
second. a = -32 (constant acceleration)
a. Find the velocity and position functions of the ball.
v = -32 dt = -32t + C  -32t + 96
a = (-32t + 96) dt = -32t2/2 + 96t + C  -16t2 + 96t + 0
b. When is the ball moving up? velocity positive
-32t + 96 = -32(t – 3) > 0  t – 3 < 0  t < 3 sec.
c. When is the ball at its maximum height? velocity = 0
t = 3 sec.
d. What is the ball’s maximum height?
h(3) = -16(3)2 + 96(3) = 144 ft.
x)dx
What is the summation formula for upper or lower sums?
n
A » å f (ci )Dx where Δx = (b – a)/n and ci = mi for lower or Mi for upper.
i=1
5. Use upper and lower sums to approximate the area of the region using four subintervals of equal
width. Δx = (2 – 0)/4 = ½
A ≈ S = (½)(10) + (½)(8) + (½)(5) + (½)(10/(1.52 + 1) ≈ 13.038 units2
A ≈ s = (½)(8) + (½)(5) + (½)(10/(1.52 + 1) + (½)(2) ≈ 8.538 units2
For the graph of f shown above, find:
5
6.
0
ò f (x)dx
8.
f (x)dx =05f(x)dx+05f(x)dx 9.
ò f (x)dx =45f(x)dx+57f(x)dx
4
0
½(1)(1) + -(½)(2)(2) = ½ – 2
= -1.5
3.5 + -(½ 4(2)) = -1.5
12. Given
a.
6
6
2
6
2
7
11.
ò
f (x) dx make f pos.
4
= ½(1)(1) + (½)(2)(2) = 2.5
ò f (x)dx = 10 and ò g(x)dx = 3 , evaluate…
ò [ f (x) + g(x)] dx = 10 + 3 = 13
6
c.
2
ò [ f (x) - g(x)] dx = -10 – (-3) = -7
6
ò [ 2 f (x) - 3g(x)] dx = 2(10) – 3(3) = 11
2
2
b.
make area pos.
= |-1.5| = 1.5
7
9
ò f (x)dx
4
1.5
½ (2 + 5)(1) = 3.5
ò
10.
9
0
7.
ò
7
f (x)dx = -09f(x)dx
6
d.
ò 5 f (x)dx = 5(10) = 50
2