Calc 2

Can you give me the step by step instructions on how to do this problem? I'm having difficulty understanding Riemann Sum.

Let f(x) = 2/x

a. Compute the Riemann sum for R4 using 4 subintervals and right endpoints for the function on the interval [1,5].

b. Draw a figure with the appropriate rectangles labeled corresponding to the Riemann sum R4.

c. Find the exact value of the integral from [1, 5] of f(x)dx = F(5) - F(1), by finding an antiderivative F(x) of f(x) = 2/x

~~This is from my assignment that I got back, which I thought I did right, but I didn't. I really want to understand the concept of Riemann sums. Thanks a lot!! :)

  1. 👍 0
  2. 👎 0
  3. 👁 202
asked by Bae
  1. the sums are an approximation of the area under a curve, using rectangles. The area of each rectangle is its width times its height. The width, naturally, is the length of the interval, divided by how may subintervals you use. The height is just the value of the function at some point on the top of the rectangle: either at the left, middle, or right.

    That's all there is to it. What did you do, to get the wrong answers? I know it's easy just to post a question and wait for the answers, but when you say you already tried and got it wrong, it's helpful to show what you did; maybe your mistake is just a simple algebra error.

    1. 👍 0
    2. 👎 0
    posted by Steve
  2. Well for part a, this would be my work:
    f(x) = 2/x
    f'(x)= 2ln|x|
    = pi(2/(pi/4) + 2/(pi/2) + 2/(3pi/4) + 2/(pi)
    = pi(2/pi + 4/pi + 8/3pi + 2/pi)
    = 2+4+8/3+2 = 8/3 + 8 = 32/3

    b. I realized my mistake with the graph. One question though, what do they mean when they say "with the appropriate rectangles labeled"

    c. I got right.

    1. 👍 0
    2. 👎 0
    posted by Bae
  3. they want you to use rectangles where the top of the rectangle is determined by the value of f(x) on the right side of the subinterval.

    For [1,5] with 4 subintervals, each subinterval is of width 1, so you rectangles will have heights 2/2, 2/3, 2/4, 2/5

    If using left sides, they would have heights 2/1, 2/2, 2/3, 2/4

    For part a, it is customary to use the names f(x) and F(x) where you are approximating the area under the curve for f(x)

    In this case, f'(x) = -2/x^2
    F(x) = ∫f(x) = 2 ln x

    What's all this pi/4 stuff? I don't see any pi needed anywhere. There are no trig functions or circles involved, and you're not rotating any section of area.

    1. 👍 0
    2. 👎 0
    posted by Steve
  4. Oh. I thought you needed to use pi in Riemann.

    Ok, so you would just add the intervals, and that would be your answer, correct?
    1+2/3+1/2+2/5 = 77/30 ?

    & the labeling, would you just label each rectangle 'R1' R2' 'R3' and 'R4' ?

    1. 👍 0
    2. 👎 0
    posted by Bae
  5. In this case, since the width is 1, you are correct. I would label the rectangles as you did. Make it clear that the curve passes through the upper right corner of each rectangle.

    A good diagram is at

    http://www2.seminolestate.edu/lvosbury/CalculusI_Folder/RiemannSumDemo.htm

    Since your curve is descending, the rectangles will all be below the curve.

    1. 👍 0
    2. 👎 0
    posted by Steve
  6. Yep I saw the diagram. Ok so the width is always one? If it asks for say 8 intervals, would it still be a width of 1?
    and the interval would be from [0,9]

    1. 👍 0
    2. 👎 0
    posted by Bae
  7. No, the width is not always 1.
    Divide the interval into the desired number of intervals.

    In this case, interval=4, number of divisions=4.

    If you want 8 intervals on [0,9], each would be 9/8 in width.

    (b-a)/n is the subinterval width to divide [a,b] into n equal pieces. Don't forget your algebra I just because you're taking calculus now!

    And, of course, the subintervals don't have to be all the same width. That's just makes the math easier. For more accurate estimates in areas where the graph is very curvy, you might want to use narrower intervals, and wider ones where the graph is relatively straight.

    1. 👍 0
    2. 👎 0
    posted by Steve
  8. Whoa, you just summed up my professor's 2 lectures on Riemann Sums into a mere 4 text posts. Thank you so much!!!

    1. 👍 0
    2. 👎 0
    posted by Bae

Respond to this Question

First Name

Your Response

Similar Questions

  1. algebra

    please give me step by step instructions on how to solve the following equations; x-y=3 and -x+4y=9 I am trying to graph the coordinates.

    asked by sherry on February 1, 2010
  2. Algebra

    How to solve this problem √363−3√27 Need step by step instructions. Thanks, Kaai97

    asked by Kaai97 on May 3, 2016
  3. Algebra

    How to solve this problem √363−3√27 Need step by step instructions. Thanks, Kaai97

    asked by Kaai97 on May 3, 2016
  4. math

    Consider the sequence of steps to solve the equation: 2(x − 4) + 6x = 9x − 10 Which step in solving this equation is justified by the Commutative Property of Addition? Given ⇒ 2(x − 4) + 6x = 9x − 10 Step 1 ⇒ 2x − 8

    asked by Anonymous on October 25, 2018
  5. Algebra 1

    How do you write the equation of a line if you have two points? Please use detail and give step by step instructions: trying to study for a test and I don't understand this concept.

    asked by Alex on February 13, 2014
  6. calculus

    h(t)=cot t a.) [pi/4,3pi/4] b.) [pi/6,pi/2] Find the average rate of change of the function over the give n interval and intervals. Please show step by step instructions. Thanks

    asked by emmanuel on August 27, 2017
  7. 8th grade

    I am having trouble with these kinds of problems and I ask my teacher but he doesnt explain it good enough. Will you please tell me step by step instructions to solve this type of problem... -2(z + 11)=6 thank you

    asked by Sara on September 23, 2008
  8. 8th grade Algebra I (junior level math)

    Hello, I am currently in Unit 4-1 (Ratio, and Proportions) and I don't understand how pairs of ratios could form proportions. Could someone give me step by step instructions? Thank You

    asked by Jay on December 1, 2009
  9. Math

    I need help solving the problem not the answer. (Multiple Choice) The steps below show the incomplete solution to find the value of x for the equation 7x + 8 − 3x = −6 + 10: Step 1: 7x + 8 − 3x = −6 + 10 Step 2: 7x + 8 −

    asked by Ryan on March 30, 2017
  10. maths - trigonometry

    I have my maths exam on Tuesday and cannot remember how to do trigonometric identities at all. I was so sure, but now I'm not. I asked a similar question earlier, but I still don't quite understand. Please give me step-by-step

    asked by anonymous on June 18, 2010

More Similar Questions