Math 121: Chapter 4 Exam Study Guide

In-Class Exam

1. Find the indefinite integral. Six parts.
2. Evaluate the definite integrals. Three parts.
3. A curve with the areas between the curve and an interval are given. Use the figure to find the definite integrals and average values. Thirteen parts.
4. Use the numeric integration feature of your calculator to evaluate the definite integrals. Two parts.
5. Rewrite the Riemann sum as a definite integral and then evaluate. For example, \(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left( {{{\left( {9 + {\textstyle{{4k} \over n}}} \right)}^2} + 3\left( {9 + {\textstyle{{4k} \over n}}} \right) - 2} \right)\left( {\frac{4}{n}} \right) = \int_9^{13} {\left( {{x^2} + 3x - 2} \right)dx} = \frac{{1840}}{3}} \).
6. Evaluate the sums. 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. Two parts.
7. 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 \(F''(\pi)\). Six parts.
8. Completely rewrite the integral and limits in terms of u. Do not evaluate the integral after rewriting. Three parts.
9. 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.

Take-Home Exam

1. Numerical integration. Find the average value using left hand endpoints, right hand endpoints, and the trapezoid method.
2. Numerical integration. Find the average value using left hand endpoints, right hand endpoints, the trapezoid method, and Simpson's method.
3. Riemann sum problem. Evaluate the limit using properties of limits. Then convert the limit into a definite integral and evaluate it that way. Show work in both cases and give exact answers.

Notes

• Problems 1 and 2 must be worked without a calculator and turned in before getting problems 3-9.
• You can verify the answer in #5 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

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Last updated October 30, 2012 10:38 PM