1. Find the following indefinite integrals:
R
(a) cos(x) dx
R
(b) sin(x) dx
R
(c) sec2 (x) dx
R
(d) sec(x) tan(x) dx
R
(e) 3 dx
R
(f) 2x dx
R
(g) x3 dx
R √
(h) 3 x dx
R
(i) x−3/4 dx
R
(j) (7x2 + 8x − 9) dx
Z
x
√
2. Integrate
dx.
1 + x2
Z
3. Integrate x · f 0 (xx ) dx.
Z
2
(x2 + 2x − 4) dx.
4. Evaluate
−1
Z
3π
4
5. Evaluate
sin(x) dx
− π4
Z
1
6. Evaluate
0
Z
x
dx.
(x + x2 )3
1
x3 cos(x4 ) dx.
7. Evaluate
−1
Z
8. Integrate
sin(3x + 1) dx.
9. In the following parts, find the area enclosed by the curves y =
√
x and y =
x+3
4
(a) Find the x-values where the curves intersect.
(b) Using an x which has a square root you can actually evaluate, say x = 4, find out
which function is larger between where they intersect.
(c) Write the appropriate definite integral that gives the area enclosed by the two
curves.
(d) Evaluate the definite integral from part (c).
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10. Let f be the function f (x) = sin2 (x), and consider the solid of revolution formed by
rotating the function about the x-axis. A single “bulb” as shown below is obtained by
letting x vary from 0 to π.
f (x) = sin2 (x)
2
1
1
x
−2 −1
1
2
3
4
0
5
−1
−1
0
−2
1
2
0.5
0
−0.5
3
(a) Write the volume of the solid from x = 0 to x = π as an integral.
(b) Find the volume of this solid by evaluating the integral.
(A trigonometric identity will be needed to evaluate this integral. The method of
u-substitution will not work as well you might hope.)
11. Let f be the function f (x) = 4x2 − x4 from x = 0 to x = 2, and consider the solid of
revolution formed by rotating the function about the y-axis.
5
f (x) = 4x2 − x4
4
3
4
2
2
1
x
−2 −1
−1
1
2
3
4
5
2
0
1
−1
−2
0
0
1
−1
2 −2
(a) Write the volume of the solid from x = 0 to x = 2 as an integral.
(b) Find the volume of this solid by evaluating the integral.
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