Name:
Date:
Calculus
Fundamental Theorem of Calculus and Integration by Substitution Test Review
What is the fundamental theorem of calculus in (your own) words?
Evaluate the following.
3p / 4
9
1.
òx
xdx
ò sinqdq
2.
4
0
4
3. Sketch the graph of the region whose area is given by
ò (x
2
- 9)dx . Then find the area.
3
4. Determine the area of the region below. (You will need a calculator.)
See p. 317 # 47, 49, 53, 57 for solutions
What is the Second Fundamental Theorem of Calculus?
Explain the Mean Value Theorem for Intervals in (your own) words.
How does the MVTI relate to average value?
5. Find the average value of f (x) =
1
over [4, 9]. Then find the value of x at which the function
x
assumes this value.
x
6. Use the Second Fundamental Theorem of Calculus to find F '(x) for F(x) =
òt
0
See p. 318 # 61 and 63 for solutions
2
1+ t 3 dx .
Name:
Date:
Calculus
7. The traffic flow rate (cars/hour) across an intersection is r(t) = 300 + 600t – 90t2, where t is in
hours, and t = 0 is 6 AM.
a. Write a function that would give you the number of cars passing through during the first x
hours.
b. How many cars pass through between 6 and 10 AM?
Find the indefinite integral.
x2
dx
8. ò
x3 + 3
9.
Evaluate the definite integral.
2
10. ò (x +1) 1 - xdx
-1
11.
ò sin
3
p
æ xö
x cos xdx
ò cosçè 2 ÷ødx
0
See p. 318 #69, 73, 81, 87 for solutions. For #7: http://www.youtube.com/watch?v=M795QnWGzwY