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Calc II - CU Boulder - Fall 2010 - Section005

Midterm 1 Info

Wednesday, September 15, 2010 - 17:15 - 18:45

Our midterm will be held in HALE270 during 5:15-6:45PM on Wednesday, September 14.

Book Assignment 4

Due Date: 
Monday, September 20, 2010 - 16:00

Section 7.8: Exercises 18, 20, 22
Section 8.1: Problem 14 and Exercise 26

Improper Integral Comparisons

Monday, September 13, 2010 - 14:00

Math 2300 Section 005 – Calculus II – Fall 2010

Comparison of Improper Integrals Examples – September 13, 2010

Comparison of Improper Integrals

  1. \int _{1}^{\infty}\frac{1}{\sqrt{x^{3}+5}}\, dx

    Solution: This is example 1 from Section 7.8 in the text.

  2. Converge or Diverge?

    \int _{{10}}^{\infty}\frac{1}{\ln(t)}\, dt

    Solution: Notice that for t\ge 10 we have


    and so

    \int _{{10}}^{\infty}\frac{1}{\ln(t)}\, dt>\int _{{10}}^{\infty}\frac{1}{t}\, dt=\infty

    and so this integral diverges. Notice that starting the integral limit at t=10 makes no difference here. The simple fact is that we know

    \int _{1}^{\infty}\frac{1}{t}\, dt=\int _{1}^{{b}}\frac{1}{t}\, dt+\int _{b}^{\infty}\frac{1}{t}\, dt

    is infinite, while the integral

    \int _{1}^{b}\frac{1}{t}\, dt

    is finite for any b satisfying 1<b<\infty. It is really the integral on the far left, the one with the infinite limit, that causes the problem here.

Improper Integrals

Wednesday, September 8, 2010 - 14:00 - Thursday, September 9, 2010 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

Improper Integrals Examples – September 8-10, 2010

  1. Limits of Integration Not Finite

    \int _{1}^{\infty}\frac{1}{x^{p}}\, dx

    Solution: We find that

    \displaystyle\int _{1}^{\infty}\frac{1}{x^{p}}\, dx \displaystyle=\lim _{{b\rightarrow\infty}}\int _{1}^{b}\frac{1}{x^{p}}\, dx
    \displaystyle(p\neq 1) \displaystyle=\lim _{{b\rightarrow\infty}}\left[\frac{1}{-p+1}x^{{-p+1}}\right]_{1}^{b}
    \displaystyle(p\neq 1) \displaystyle=\frac{1}{-p+1}\left(\lim _{{b\rightarrow\infty}}b^{{-p+1}}-1\right).

    This will converge if and only if p>1, meaning that it will diverge when p\le 1. We showed this in class in somewhat painful detail.

Midterm 1 Review

Math 2300 – Calculus II – University of Colorado

Fall 2010 – Review for Midterm I

One of the main areas that will be emphasized during this midterm, compared to previous calculus II exams that you may find, is abstract and conceptual understanding. From what we have done in class and on homework/webwork, you have all of the tools to solve these problems. But, they may initially look a bit unfamiliar. Some examples of such problems are given here to help prepare you for the sorts of questions that will be asked on the midterm.

Some Things to Make Sure You Know

  • Substitution of indefinite and definite integrals.

  • Integration by parts of indefinite and definite integrals. It is also always helpful conceptually to know how to derive the formula for integration by parts.

  • Know how to use the table of integrals. More importantly, know how to derive some of the “basic” formulas on and related to that table: \int e^{x}\cos x\, dx and \int\cos^{n}x\, dx are classic examples, and you did one of them as a homework problem.

  • Know how to use the method of partial fractions, including how to apply long division to simplify an integral.

  • Know which trig substitutions to use, and how to use completing the square to rewrite quadratics when needed.

  • Understand how the Left, Right, Mid and Trap approximation methods work and how they relate (when a function is increasing/decreasing, concave up/concave down). Know how Simp relates. Of course, you should be able to calculate any of these and produce the relevant formulae for their calculation (even if you are using a calculator). Know what increase in the factor of subdivisions results in one more decimal of accuracy for each method.

  • Know how to set up the calculations for improper integrals, be able to follow through with the corresponding limit calculations, and be able to explain why this weird limit process is needed.

Take Home Quiz - Improper Integrals

Due Date: 
Monday, September 13, 2010 - 16:00

MATH2300-005 – Fall 2010 – University of Colorado

Quiz - Due Monday, September 13, 2010

  1. Evaluate the integral

    \int _{1}^{4}\frac{dx}{(x-2)^{{2/3}}}\, dx
  2. Evaluate the integral

    \int _{{-\infty}}^{0}\frac{e^{x}\, dx}{3-2e^{x}}
  3. Replacement Question: While trying to escape a wildfire, a family of four prairie dogs comes to a bridge. They realize that the bridge is old and only two of them may cross at a single time. Making matters worse, it is night and they only have one headlamp. So, two will cross and then one will return with the headlamp, then two more will cross and one will return, etc. When two prairie dogs cross the bridge, they will always travel at the pace of the slower prairie dog. Daddy Prarie Dog can cross the bridge in 1 minute, Mommy Prairie Dog can cross in 2 minutes, Junior Prairie Dog crosses in 5 minutes and Baby Prairie Dog crosses in 10 minutes. If the fire will arrive at their location and burn the bridge, taking with it any possibility of escape, in 17 minutes and 30 seconds, is there a chance for the family to survive?

Worksheet 3

Solutions to worksheet 3 are posted.

Book Assignment 3

Due Date: 
Monday, September 13, 2010 - 16:00

Section 7.6: Problem 8
Section 7.7: Problems 28 and 50

Integral Approximations

Tuesday, September 7, 2010 - 14:00

Math 2300 Section 005 – Calculus II – Fall 2010

Approximation Notes and Examples – September 3 & 7, 2010

Main Idea: If we are trying to approximate the integral

\int _{a}^{b}f(x)\, dx

and we are subdividing the interval [a,b] into n\in\mathbb{Z}^{+} (n is a positive integer) subintervals, then we have five methods at our disposal. The first three are geometrically defined (LEFT, RIGHT and TRAP), while the fourth and fifth are a modification of the other methods (MID and SIMP). The bottom diagram shows the difference in LEFT, RIGHT, TRAP and MID when f(x)=\sin(x)+1, [a,b]=[1,5] and n=4.

tikz graph

Partial Fractions Quiz

MATH2300-005 – Fall 2010 – University of Colorado

Quiz - Friday, September 3, 2010

  1. Explain how you know that an integration problem might involve integration by parts. That is, what clues you in to the idea that integration by parts might be a valid approach to solving an integral?

    Solution: The main thing I am looking for here is something along the lines of the following: The integrand appears to be a product of two functions u and v^{{\prime}}, where we can find u^{{\prime}} and v and the integral of u^{{\prime}}v is in some way easier than the original integral. But, there are other instances when integration by parts comes up that don't look like this, such as when integrating \ln x.

  2. Solve the integral, showing all calculations that lead to your result. (You may use a calculator, but your calculations need to be capable of being followed as if the integral is computed by hand.)

    \int\frac{3}{(x-2)(x-5)}\, dx

    Solution: This is an obvious partial fractions problem. First, write

    \int\frac{3}{(x-2)(x-5)}\, dx=3\int\frac{1}{(x-2)(x-5)}\, dx.

    and then compute the partial fractions.


    Multiplying on both sides by (x-2)(x-5) and then putting the polynomial on the right in standard form gives

    \displaystyle 1 \displaystyle=A(x-5)+B(x-2)
    \displaystyle 1 \displaystyle=(A+B)x-5A-2B

    This results in the system of equations:

    \displaystyle A+B \displaystyle=0
    \displaystyle-5A-2B \displaystyle=1

    which has the solution A=-\frac{1}{3} and B=\frac{1}{3}. So, we have

    \displaystyle 3\int\frac{1}{(x-2)(x-5)}\, dx \displaystyle=3\int\frac{-1/3}{x-2}+\frac{1/3}{x-5}\, dx
    \displaystyle=-\int\frac{1}{x-2}\, dx+\int\frac{1}{x-5}\, dx

© 2011 Jason B. Hill. All Rights Reserved.