# Math 121: Chapter 4 Exam Study Guide

- Find the indefinite integral; show work. Five parts.
- Evaluate the definite integrals. Show work and give exact answers. Three parts.
- Write the expression in summation notation. Example
\(\frac 1 2 + \frac 2 3 + \frac 3 4 + \frac 4 5 + \cdots + \frac {19}{20} =
\sum\limits_{k = 1}^{19} {\frac {k}{k+1}} \)
. You do not need to evaluate the sum, just write it in summation notation. Two parts.
- You are given the closed forms for the sums of one, the sum of the integers, and the sum of the squares of the integers. Use it to find a sum for a value of n, the general closed form, and the limit as n approaches infinity of an expression.
- Given the values of two definite integrals, use them to find other definite integrals. Five parts.
- Evaluate the limit of a sum by first writing it as a definite integral. For example,
\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left(
{{{\left( {9 + {\textstyle{{4k} \over n}}} \right)}^2} + 3\left( {9 +
{\textstyle{{4k} \over n}}} \right) - 2} \right)\left( {\frac{4}{n}}
\right) = \int_9^{13} {\left( {{x^2} + 3x - 2} \right)dx} =
\frac{{1840}}{3}} \).
- A region is shown. Find the area using geometric formulas. Then write the area as the sum of definite integrals. Do not evaluate the integrals.
- Given a function where the upper limit is defined in terms of a
variable/function, find the indicated values. Example:
Let \(F(x)=\int_0^x \sin t\, dt\)
. Find \(F(0)\), \(F(\pi)\), \(F'(0)\), \(F'(\pi)\), \(F''(0)\), and \(F''(\pi)\)
- Use the second fundamental theorem of calculus to find the derivative. Two parts.
- Rewrite the integral and limits in terms of
*u*. Do not evaluate the integral after rewriting. Two parts.
- Numerical integration -- take home.
- Numerical integration -- take home.

## Notes

- Most of the problems are similar to those in the textbook.
- Don't spend too long on any one problem. If you get stuck, make a note to come back and then move on.
- You can verify the answer in #6 with Maxima using this command: limit(nusum(((9+4*k/n)^2+3*(9+4*k/n)-2)*(4/n),k,1,n),n,inf); See the Calculus Wiki for an explanation and another example.
- There is a take home exam worth 20 points that is due the day of the in-class test.

## Points per problem

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
TH1 |
TH2 |
Total |

Pts |
20 |
15 |
4 |
9 |
5 |
5 |
6 |
6 |
4 |
6 |
10 |
10 |
100 |