Math 3B Midterm Review
Written by Victoria Kala
[email protected]
SH 6432u Office Hours: R 9:45 - 10:45am
The midterm will cover the sections for which you have received homework and feedback (Sections
4.9 - 6.5 in your book). No calculators will be allowed on the midterm. No notes or notecards will
be allowed on the midterm.
You will be expected to know formulas and how to use them, and I have provided most of them
in this review. I have also selected practice problems that should be similar to problems you have
seen in your homework. I highly recommend completing your homework before the midterm.
Here are some things you should know how to do:
• Know how to find the general antiderivative of a function
• Know how to estimate the area under a curve using left and right endpoints (or the upper
and lower estimates)
• Know how to estimate the area under a curve using midpoints
• Know how to write and evaluate a Riemann sum
• Know how to convert a Riemann sum into an integral
• Know how to evaluate an integral by interpreting it as an area
• Know how to take the derivative of an integral (FTOC I)
• Know how to evaluate a definite integral (FTOC II)
• Know how to evaluate an indefinite integral
• Know how to evaluate an integral using a substitution
• Know how to find the area between two curves
• Know how to find the volume of a solid using the disk or washer method
• Know how to find the volume of a solid using the cylindrical shells method
5.1 - Areas and Distances
We can estimate areas under curves by subdividing the area into small rectangles.
The approximate area of a function f (x) on the interval [a, b] is given by:
A≈
n
X
f (xi )∆x = (f (x1 ) + f (x2 ) + ... + f (xn ))∆x
i=1
where ∆x =
b−a
n
and xi = a + i∆x.
1
5.2 - The Definite Integral
The Riemann sum is given by:
A = lim
n→∞
n
X
b
Z
f (xi )∆x =
f (x) dx
a
i=1
Sometimes we can evaluate the Riemann sum without converting to an integral. Here are some
helpful formulae (you are expected to know these for the test!):
n
X
1=n
i=1
n
X
i=
i=1
n
X
i2 =
i=1
n
X
n(n + 1)(2n + 1)
6
i3 =
i=1
n(n + 1)
2
n2 (n + 1)
4
2
Example: Find the exact area under f (x) = x3 + x2 + x + 1 on the interval [0, 1].
First, we want to find ∆x and xi :
∆x =
1−0
1
b−a
=
=
n
n
n
xi = a + i∆x = 0 + i
1
i
=
n
n
Then we plug these into the above formula:
A = lim
n→∞
n
X
i=1
f (xi )∆x = lim
n→∞
n
X
xi 3 + xi 2 + xi + 1 · ∆x
i=1
!
3 2
n
X
i
i
i
1
= lim
+
+ +1 ·
n→∞
n
n
n
n
i=1
n 3
X
i
i2
i
1
= lim
+ 2 + +1 ·
3
n→∞
n
n
n
n
i=1
!
n
n
n
n
X
X
X
i3
1
i2
i X
= lim
+
1 ·
+
+
3
2
n→∞
n
n
n
n
i=1
i=1
i=1
i=1
!
n
n
n
n
X
1 X 3
1 X 2 1X
1
= lim
i + 2
i +
i+
1 ·
3
n→∞
n i=1
n i=1
n i=1
n
i=1
2
!
2
1 n2 (n + 1)
1 n(n + 1)(2n + 1)
1 n(n + 1)
1
·
= lim
+ 2·
+ ·
+n ·
n→∞
n3
4
n
6
n
2
n
!
2
n2 (n + 1)
n(n + 1)(2n + 1) n(n + 1)
1
= lim
+
+
+n ·
n→∞
4n3
6n2
2n
n
!
2
n(n + 1)(2n + 1) n(n + 1) n
n2 (n + 1)
+
+
+
= lim
n→∞
4n4
6n3
2n2
n
4
n + ... 2n3 + ... n2 + ... n
+
+
+
= lim
n→∞
4n4
6n3
2n2
n
4
3
2
...
2n
...
n
...
n
n
+
+
+
+
+
+
= lim
n→∞ 4n4
4n4
6n3
6n3
2n2
2n2
n
1
1
1
...
...
...
1
+ 4+ + 3+ + 2+
= lim
n→∞ 4
4n
3 6n
2 2n
1
1 1 1
25
= + + +1=
4 3 2
12
The above shows the work and final answer. We can double check our answer by using an integral:
4
Z 1
Z 1
x3
x2
1 1 1
25
x
1
f (x) dx =
+
+
+x = + + +1=
(x3 + x2 + x + 1) dx =
4
3
2
4
3
2
12
0
0
0
5.3 - The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus has two parts:
R g(x)
I. If F (x) = h(x) f (t) dt then F 0 (x) = f (g(x))g 0 (x) − f (h(x))h0 (x).
II.
Rb
a
f (x) dx = F (b) − F (a) where F is the antiderivative of f .
5.4 - Indefinite Integrals
An indefinite integral returns the general antiderivative:
Z
f (x)dx = F (x) + C
where F is the antiderivative of f .
5.5 - The Substitution Rule
If u = g(x), then du = g 0 (x)dx, so we have:
Z b
Z
f (g(x))g 0 (x) dx =
a
g(b)
f (u) du
g(a)
Usually u (your substitution variable) is inside of another function.
3
6.1 - Areas Between Curves
For two functions f (x) and g(x) on [a, b], the area between them on that interval is given by:
Z b
|f (x) − g(x)|dx
A=
a
R
Another way to remember this that A = (top − bottom)dx.
We can also integrate over “functions” of y. The area on the interval y = c to y = d is given by
Z d
A=
(right − left)dx
c
6.2 - Volumes (Disk or Washer Method)
The volume of a solid S with cross-sectional area A perpendicular to the x-axis is given by:
Z
V = A(x)dx
The volume of a solid S with cross-sectional area A perpendicular to the y-axis is given by:
Z
V = A(y)dy
2
2
Usually A = π(radius)2 , or, if you have two curves, A = π(outer radius) − π(inner radius) .
Example (6.2 Example 6, pg 436) Find the volume of the solid by rotating the region found by
√
x = y and x = y about the line x = −1.
The best thing to do would be to draw a graph. The cross-sectional area is perpendicular to
the y-axis, so we will be integrating with respect to y.
x = −1 is going to be the center of our solid, so the outer radius is given by 1 +
√
inner radius is given by 1 + y. Then A = π(1 + y)2 − π(1 + y)2 , so
Z 1
π
√
V =
π(1 + y)2 − π(1 + y)2 dy = ... =
2
0
6.3 - Volumes by Cylindrical Shells
The volume of a solid rotated about the y-axis between two curves is given by
Z
V = 2πx(top − bottom)dx
About the x-axis between two curves is given by
Z
V = 2πy(right − left)dy
4
√
y and the
Example Find the volume of the solid obtained by rotating about the y-axis the region between
y = x and y = x2 .
A quick graph shows that y = x is the top function and y = x2 is the bottom function on the
region from x = 0 to x = 1. Since we are rotating about the y-axis, then we integrate with respect
to x. Thus the volume is given by:
Z 1
π
V =
2πx(x − x2 )dx = ... =
6
0
6.5 - Average Value of a Function
The average value of a function f (x) on the interval [a, b] is given by
fave =
1
b−a
Z
b
f (x)dx
a
Study Tips
• Complete all your homework
• Study important formulas using flashcards
• Review notes and homework
• “Data Dump”: when you enter the test, write all the formulas you know from memory on the
margin of your test. This way you won’t have to remember those formulas during the middle
of the exam.
• Get a good night sleep and don’t take the test on an empty stomach!
Practice Problems
The following are problems from your textbook. These problems should be similar to ones you have
seen on the homework, and should be a good review for the midterm.
1. (4.9 #46) Find f given f 00 (t) = 2et + 3 sin t, f (0) = 0, f (π) = 0.
√
2. (4.9 #60) Find the position of a particle given the following data: v(t) = 1.5 t, s(4) = 10.
3. (5.1 #7) Find the upper and lower sums for f (x) = 2 + sin x, 0 ≤ x ≤ π with n = 2, 4, 8.
R3
4. (5.2 Example 2, pg 375) Use the Riemann sum to evaluate 0 (x3 − 6x) dx.
5. (5.2 #17) Express the following limit as a definite integral on the interval [0, 6]:
lim
n→∞
n
X
xi ln(1 + x2i )∆x
i=1
5
6. (5.2 #38) Evaluate the integral by interpreting it in terms of areas:
7. Find the derivative of
√
Z
h(x) =
ex
x
R5
(x −
−5
√
25 − x2 ) dx
z2
dz
sin z + 2
R1
8. (5.3 #37) Evaluate 0 (xe + ex ) dx.
9. (5.3 #42) Evaluate
R 1/√2
1/2
√ 4
1−x2
dx. Hint: (sin−1 x)0 =
10. Evaluate
√ 1
1−x2
√
2t2 + t2 t − 1
dt
t2
Z
11. (5.4 #18) Evaluate
Z
12. (5.4 #40) Evaluate
2ex
dx
sinh x + cosh x
Z
Hint: sinh x =
ex −e−x
, cosh x
2
=
sin 2x
dx
sin x
ex +e−x
2
13. (5.5 Example 3, pg 409) Evaluate
Z
14. Evaluate
R
tan xdx. Hint: tan x =
√
x
dx
1 − 4x2
sin x
cos x .
15. (5.5 #39) Evaluate
Z
sin 2x
dx
1 + cos2 x
16. (6.1 #4) Find the area between x = y 2 − 4y and x = 2y − y 2 .
17. (6.1 #19) Find the area between y = cos πx and y = 4x2 − 1.
18. (6.2 #12) Find the volume by rotating the region bounded by the given curves about the
specified line: y = e−x , y = 1, x = 2 about y = 2.
19. (6.2 #17) Find the volume by rotating the region bounded by the given curves about the
specified line: x = y 2 , x = 1 − y 2 about x = 3.
20. (6.3 #6) Use the method of cylindrical shells to find the volume generated by rotating the
following region about the y-axis: y = 4x − x2 , y = x.
21. (6.3 #14) Use the method of cylindrical shells to find the volume generated by rotating the
following region about the x-axis: x + y = 3, x = 4 − (y − 1)2 .
√
22. Find the average value of the function on the given interval: g(x) = x2 1 + x3 , [0, 2].
6