Calculus
Review 4.3, 4.2, 4.6
Name:
Date:
Use the properties of integrals to determine each:
0
1. Given:
∫
4
∫ f ( x)dx = −7
f ( x)dx = 6
−2
0
Find:
0
∫
a.)
f (x) dx =
0
∫ 3 f ( x)dx =
b.)
−2
4
−2
4
c.)
∫
f ( x)dx =
d.)
∫ f ( x)dx =
−2
−2
Evaluate. Draw the graph to determine the area.
4
3
∫ ( − | x | +4) dx =
2.
∫ (5 − x ) dx =
3.
−3
0
Use the diagram below to answer
#4 – 9.
6
(10, 4))
4
(-5, 3)
(0, 3)
2
-5
5
(8, 0)
10
-2
(6, -2)
−3
4. ∫ f (x)dx
−5
3
7. ∫ f (x)dx
8
3
5. ∫ f (x)dx
−5
3
8. ∫ 4 f (x)dx
−3
3
6. ∫ f (x)dx
3
10
9. ∫ f (x)dx
−5
For each problem, sketch the graph, label the x-axis appropriately, and approximate the area using each
of the estimates listed below. Be sure to write out the set up for each problem. Retain all calculated
values to 5 decimal places and round your final answer to 3 decimal places. Use n = 4.
10. f (x) = x 3 − 8x 2 + 16x
[ 0, 4 ]
Lower Sum = ____________
Upper Sum = ____________
Midpoint Sum = ___________
Trapezoid Rule = __________
Left Sum = ______________
Right Sum = ____________
11. f (x) = 1 + x 3
[ 0, 2 ]
Lower Sum = ____________
Upper Sum = ____________
Midpoint Sum = ___________
Trapezoid Rule = __________
Left Sum = ______________
Right Sum = ____________
12. f (x) = 10 cos
x
4
[ 0, 2π ]
Lower Sum = ____________
Upper Sum = ____________
Midpoint Sum = ___________
Trapezoid Rule = __________
Left Sum = ______________
Right Sum = ____________
13. Use the Trapezoidal Rule to estimate the number of square meters of land in a lot where x and y are
measured in meters. The land is bounded by a stream and two straight roads that meet at right angles.
(You do not need a diagram to solve this!) USE 5 subintervals.
X
Y
0
125
100
125
200
120
300
112
400
90
500
90
600
95
700
88
800
75
900
35
1000
30
Trapezoid Rule = ____________________