Exercises: Integration
1. The following figure shows the graph of a function f (x).
Z 4
9.
sin(x) dx, 100 rectangles
0
y
35
10. The following figure shows the graph of a function f (x).
30
y
25
3
20
15
2
10
1
5
0
2
Z 10
4
6
x
8
0
f (x) dx as accurately as you can.
Estimate the value of
1
2
Z 3
Z 3
2.
sin
√ x dx
Z 2
3.
−2
1
9
dx
3 + x2
4
x
5
Use this graph to evaluate each of the following integrals.
0
2–3 Use five rectangles and the Midpoint Rule to estimate the
value of the given integral. Round your answers to three decimal
places.
3
Z 2
f (x) dx
(a)
2
11–14
Z 6
f (x) dx
(b)
0
0
Evaluate the given integral by interpreting it as an area.
Z 3
11.
f (x) dx
(c)
Z 5
2 dx
12.
0
(10 − 2x) dx
0
Z 0.5
4. Use the following data to estimate
f (x) dx.
Z 5
0
13.
x dx
14.
1
x
0.0
0.1
0.2
0.3
0.4
0.5
f (x)
3.6
2.6
1.8
1.2
0.8
0.6
Z 4
f (x) dx.
15.
2
2.1
2.3
2.5
2.7
2.9
f (x)
0.6
0.8
1.1
1.5
2.0
6–7 Write the integral that can be approximated using the given
Riemann sum.
√
√
√
7. 3.02 (0.04) + 3.06 (0.04) + · · · + 4.98 (0.04)
8–11
Write a Riemann sum that approximates the given integral
using the given number of rectangles.
Z 3
2
x2 dx, 50 rectangles
Z 3
16.
Z 1
−2
(x3 + 5x + 2) dx
Z π/6
18.
sin x dx
20.
0
Z 4
2 dx
22.
0
ex dx
Z 4
0
dx
x−2 dx
Z π/4
24.
0
25.
x
1
Z ln 3
23.
Z 3
1
1
Z 6
21.
cos x dx
0
Z π
19.
(x2 + 1) dx
0
17.
6. (5.01)3 (0.02) + (5.03)3 (0.02) + · · · + (5.99)3 (0.02)
8.
3x2 dx
2
x
9 − x2 dx
0
15–26
Use the Fundamental Theorem of Calculus to evaluate the
given integral.
Z 3
5. Use the following data to estimate
Z 3p
sec2 x dx
0
x3/2 dx
Z 1
26.
0
1
dx
1 + x2
Answers
Z 6
1. 256.25 (any answer between 250 and 260 is fine)
2. 1.930
3. 8.946
4. 0.85
5. 1.2
6.
x3 dx
5
8. (2.01)2 (0.02) + (2.03)2 (0.02) + · · · + (2.99)2 (0.02)
10. (a) 3
(b) 3
20. ln(3)
21. 12
(c) 14
11. 6
22. 3/4
12. 25
23. 2
24. 1
13. 12
7.
Z 5
√
x dx
3
9. sin(0.02)(0.04) + sin(0.06)(0.04) + · · · + sin(3.98)(0.04)
14. 9π/4
25. 64/5
15. 56
26. π/4
16. 12
17. −21/4
18. 1/2
19. 2