The document provides a review for a math 225 final exam covering topics including optimization, anti-derivatives, definite and indefinite integrals, area between curves, volumes, and average value of a function. Examples and problems are provided for each topic, such as finding the dimensions of a rectangle with the largest area given a perimeter of 100m or finding the average value of the function f(x)=(x-3)2 on the interval [2,5] and the value of c such that faverage=f(c).

1. MATH 225 FINAL REVIEW
5.13.2016
3.7 Optimization (EMILY) Page 250
Optimization allows you to minimize or maximize real life situations. To solve an optimization
problem, you must understand what they’re asking to start. Drawing a diagram and identifying the
given quantities and the quantities they’re asking for will help you solve this. You must assign a
symbol to the quantity being maximized/minimized and express that in terms of the other symbols.
Then, find the ​absolute​ maximum or minimum value of your function.
Example​ : A farmer has 2400 feet of fencing and wants to fence off a rectangular field that borders a
straight river. He needs no fence along the river. What are the dimensions of the field that has the
largest area?
Maximize or minimize
1. A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then
divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so
as to minimize the cost of the fence?
2. A box with a square base and open top must have a volume of 32,000 cm​3 ​
. Find the
dimensions of the box that minimize the amount of material used.
3. Find the dimensions of a rectangle with perimeter 100m whose area is as large as possible.
3.9 Anti-Derivatives (TORI)

2. Example: Find the general antiderivative of .(x)f = x√x
Find the general antiderivatives of the following:
1. Find f if f’’(x) = , f(0)=4 and f(1)=1.2x x1 2 + 6 − 4
2. Find f when f’(x) = 1 + .3√x
3. Find f when f’’(x)= 0x 2x x2 3 − 1 2 + 6
Appendix E and 4.1 Summation Notation and Areas & Distances (TORI)
Example: Use rectangles to estimate the area under the parabola from 0 to 1!y = x2
1. The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second
intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these
three seconds.
t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0
v (ft/s) 0 6.7 11.2 14.1 18.8 19.4 20
2. Use the definition to find an expression for the area under the graph of ​f​ as a limit. Do not evaluate the limit.
, 3 ≤ ​x ≤ 5(x)f = x2 +√1 x+2
3. Determine a region whose area is equal to the given limit. Do not evaluate the limit.
(9 )lim
n→∞
∑
n
i=1
n
2
+ n
2i 8
4.2 and 4.3 Definite Integrals and Fundamental Theorem of Calculus (TORI)

3. Example: Evaluate the Riemann Sum for , taking the sample points to be right(x) x x)dxf = ( 3 − 6
endpoints and a=0, b=3, and n=6.

4. 1. Set up an expression for as a limit of sums.dx∫
5
x
x4
2. Evaluate (4 x )dx∫
1
0
+ 3 2
3. Evaluate dx∫
1
0
√1 − x2
Sections 4.4 and 4.5: Indefinite Integrals and U-Substitution (EMILY) Page 321 & Page
330
​= F(x) means F’(x)=f(x).
A definite integral is a number, whereas an indefinite integral is a function. The most
general antiderivative on a given interval is obtained by adding a constant to a particular
antiderivative.
Sometimes, the FTC doesn’t tell us how to evaluate certain tricky integrals. In order to compute these,
we introduce a new variable (most of the time, we introduce u). Simply let u equal a part of your
integral, compute du, and interchange x with u and dx with du. One thing to remember is to change
your limits of integration if you are dealing with definite integrals and u-substitution.
Example: Find the general indefinite integral:
x​3​
cos(x​4​
+2) dx

5. 1. Find the integral of sqrt(2x+1) dx.
2. Find the integral of (sqrt(1+x​2​
))(x​5​
) dx.
3. Find the integral of 1/((3-5x)​2​
) dx.
Sections 5.1 (Page 344) Area Between Curves (EMILY)
In 5.1, we use integration to find areas of regions that lie between the graphs of two functions. The
area A of the region bounded by the curves y=f(x) and y=g(x) and the lines x=a, x=b, where f and g are
continuous and f(x) is greater than or equal to g(x) for all x in [a,b], is
A=the integral from a to b of [f(x)-g(x}] dx.
Example: Find the area of the region enclosed by the parabolas y=x​2​
and y=2x-x​2​
.

6. 1. Find the area of the region bounded by y=sinx, y=cosx, x=0, x=pi/2
2. Find the area of the region bounded by x=2y​2​
and x=4+y​2​
.
Section 7.1 Integration by Parts (EMILY)
Above is the general formula to integrate by parts. Your job will be to pick out when to use integration
by parts, and how exactly to use it. When you see a product of two functions that needs to be
integrated, but you cannot use u-substitution or any other integration formula, you may need to try
integration by parts. What you need to do is choose a u and a dv. The u should be something you can
differentiate easily, while the dv should be something you can integrate easily. You must find a du and
a v (by differentiating and integrating, respectfully) and then plugging it into the formula above.
Example:
Find the integral of :
te−​7​t
​dt

7. 1. Integrate: (x​2​
+ 4x)cos(x) dx
2. Integrate: t​3​
+ln(t) dt
3. Integrate: e​x​
sinx dx
Sections 5.2 and 5.3: Volumes (disks/washers and cylindrical shells) (TORI)
Example: Find the volume using cylindrical of the solid obtained by rotating about the y-axis the
region between y=x and .y = x2

8. 1. Find the volume of the solid obtained by rotating about the x-axis the region under the curve
from 0 to 1.y = √x
2. Find the volume of the solid obtained by rotating about the y-axis the region by .xy = 2 2 − x3
Section 5.5: Average value of a function (EMILY)
Above is the formula to find the average value of f on the interval [a, b]. If f is continuous on [a, b]
then there exists a number ​c in [a, b] such that:
Essentially, for positive functions f, there is a number c such that the rectangle with the base [a,b] and
height f(x) has the same area as the region under the graph of f from a to b.
Example: Find the average value of f on the given interval, and find x such that f​average​ = f(c).
f(x)=(x-3)​2​
, [2,5]

9. 1. Find the average value of the function f on the given interval and find c such that f​ave​=f(c):
a. f(x)=x​1/2 ​
, [0,4]
b. f(x)= 2x / (1+x​2​
)​2