Stony Brook University
Mathematics Department
Julia Viro
Calculus II
MAT 132
Fall 2016
Practice Midterm 1
Midterm 1 will cover definite integrals, integration techniques, improper integrals, area and
volumes by integrals (Sections 5.1-5.7, 5.10, 6.1-6.3, Appendix H2 of the textbook). Problems below
are similar to the problems you may encounter on Midterm 1. For your convenience, the problems are
rubricated. Make sure that you can handle problems from all rubrics.
Definite integral as area
1. Without performing integration, explain why
π
<
3
Zπ/2
π
dx
<
sin x
2
π/6
2
2. The function f (x) = ex is not integrable in elementary functions (it doesn’t have an elementary
antiderivative). Use geometric interpretation of definite integral in order to find upper and lower
Z1
2
bounds for ex dx, that is find numbers a, b such that
0
Z1
a<
2
ex dx < b
0
√
2

 1−x ,
3. Evaluate
f (x) dx, where f (x) = 2,

x − 2,
0
the interval [0, 3]?
Z3
Z5 p
4. Evaluate the integral
4 − (x − 3)2 dx
1
0≤x≤1
1 < x ≤ 2 . What is the average value of f over
2 < x ≤ 3.
2
Definite integral as limit of Riemann sums
Z3
5. Evaluate the integral
(2x + 3) dx as the limit of a Riemann sum (take subintervals of equal
−1
lengths; use either left, or right, or middle endpoints). You may use the formula
n
P
i=
i=1
6. Use definite integrals to calculate the following limits
12 + 22 + · · · + n2
1
1
1
a) lim
+
+ ··· +
b) lim
n→∞
n→∞
n3
n+1 n+2
n+n
Fundamental theorem of calculus
Zt
7. Let f (t) =
ln3 x dx. Find f 0 (3).
e
Zx2
8. Let f (x) =
2
eu du. Find f 0 (x).
2x
Zx
9. Let f (x) =
2
te−t dt, where x ≥ 0. Find local extrema and inflection points of f (x).
0
10. Evaluate the following integrals
Z2
Z1
√ x
√
2 + x2
√
dx
a) ( x − ( 2) ) dx b)
3
x
Zπ/4
c)
1
0
Z1
11. Calculate
1
dx. Is your answer plausible?
x2
−1
Z4
12. Calculate
−3
|x2 − x − 6| dx
π/6
sin 2x
dx
sin x
Z1
d)
−1
|4 − x2 |
dx
x−2
n(n + 1)
.
2
3
Integration techniques
Zπ/2
Zπ/2
13. Prove that if f (x) is continuous on [0, 1], then
f (sin x) dx =
f (cos x) dx.
0
0
14. Evaluate the following integrals
3x2 − 2x + 1
dx
x2 − 2x + 1
Z
a)
Z
b)
Z
arccos x dx
c)
Z1
x(x2 + 3)5 dx
d)
x3
dx
x4 + 1
−1
√
ex − 1 dx
2
3x − x + 1
dx g) e2x sin 3x dx h)
3
2
x +x
√
Z
Z
Ze2
x−1
sin x
dx
√
dx
j)
i)
dx
k)
l)
2
x − 2x + 5
x ln x
x
Z
Z e
Z
Z
dx
x+1
2
3
√
√
p)
dx o)
m)
cos x sin x dx n)
2 − 3x
1 − x2
Z
Z √
Zπ/6 2
cos x − sin2 x
q) cos2 x sin2 x dx r)
4 − x2 dx s)
dx
sin 2x
Z
e)
Z
Z
Z
f)
x2
dx
√
dx
1 + x2
Z
cos(ln x) dx
x−8
dx
−x−6
Z
t) x2 e−x dx
x2
π/8
Zπ/4
u)
x3 tan2 x dx
−π/4
Improper integrals
15. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
Z1
a)
dx
2x − 1
Z∞
b)
−1
Z∞ √
3
e)
ln x
dx
x
c)
−1
1
Z∞
f)
dx
2
x +4
−∞
1
Z1
dx
2x − 1
Z∞
g)
dx
p
3
(2x − 1)2
dx
x ln x
e
Zπ/2
h)
tan x dx
0
Z∞
d)
1
Z∞
i)
−∞
16. State whether the integral converges or diverges and justify your claim.
Z∞
a)
0
x2
dx
1 + x5
Zπ
b)
0
sin x
dx
x
Zπ/2
c)
−π/2
dx
sin x
dx
p
3
(2x − 1)2
x2
dx
1 + x6
4
Mean Value Theorem for Integrals
√
17. Find the average value of the function f (x) = a2 − x2 over the interval [−a, a]. Draw a picture
and explain the geometrical meaning of the average value.
18. Find the value of a for which the average value of the function f (x) = ln x over the interval [1, a]
is equal to the average rate of change of f (x) over this interval.
Area of plane regions enclosed by curves
19. Find the area of the region bounded by the curves y = x3 and y =
integrating along the x-slices and by integrating along the y-slices.
√
x in two different ways: by
20. Find the area of the region that lies above y = 0, to the right of x = 1 and under the graph of
1
f (x) = 2
.
2x + 5x + 2
(
x(t) = 1 + et
21. Find the area of the region enclosed by the x-axis and the curve
y(t) = t − t2 .
22. Find the area of the polar region enclosed by the smaller loop of the curve r = 1 + 2 cos θ.
Volumes of solids of revolution
23. The finite region bounded by the curves y = tan x, y = 0, x = π/4 is rotated about a) the x-axis
b) the line y = −1. Find the volumes of the solids of revolution.
24. The finite region bounded by y = x2 , x = 0, and y = 1 is rotated about the y-axis. Find the
volume of solid of revolution in two different ways: by slicing and by cylindrical shells.
25. The region is situated below the curve y = e−x , above the x-axis, and to the right of y-axis. Find
the volume of the solid of revolution if the region is rotated a) about the x-axis and b) about the
y-axis.
Answer Key (typos may occur)
1. Use geometric interpretation of the integral as area. Estimate the integrand.
2. Answer may vary. Use geometric interpretation of the integral as area.
3.
π + 10
,
4
4. 2π
5. 20
π + 10
12
5
6. a)
1
3
b) ln 2
7. ln3 3
4
8. 2xex − 2e4x
2
√
9. x = 0 is a local minimum, x =
√
2 ln 2 + 6 − 6 2
10. a)
3 ln 2
b)
27
8
2
is an inflection point
2
c)
√
2−1
d) −4
11. −2 is an incorrect answer.
12.
53
2
13. Hint: sin x = cos
π
2
−x
14. You can check your answer by differentiation or use some online integral calculators (like Wolfram).
√
3+333
π
π
15. a) div b) div c)
d) div e) div f )
g) div h) div i)
2
2
3
16. a) conv
17.
b) conv
c) div
πa
. Don’t forget to draw a picture!
4
18. e
19.
5
12
20.
ln 2
3
21. 3 − e
√
2π − 3 3
22.
2
π
23. a) π 1 −
4
24.
π
2
25. 2π
π
b) π 1 − + ln 2
4