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Math Help - 6 problems! Any help appreciated :)

  1. #1
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    6 problems! Any help appreciated :)

    So my professor (I'm in college, freshman) sent us an online study sheet with 6 problems and I just need some help solving them, please, any answers and explanations would be appreciated!...I'm not sure about any of my answers.. I'm just having so much trouble with this! I keep thinking I'm getting the answer, I'll come up with something, and then it won't even be close to the answers. Or it'll be close but all the answers are within a tenth of each-other! It's driving me crazy because I'm so scared now that I won't pass the next test! Please help, explanations, answers, anything!


    1. Use midpoints to approximate the area under the curve (see link) on the interval [0,1] using 10 equal subdivisions.

    http://imagizer.imageshack.us/v2/800x600q90/707/5b9m.jpg

    3.157---my answer (but I don't understand midpoints)
    3.196
    3.407
    2.078
    2.780


    2. Use right-hand endpoints and 6 equal subdivisions to approximate the area beneath the curve on the interval [0, 6].

    http://imagizer.imageshack.us/v2/800x600q90/38/7ruq.jpg

    0.9243
    1.405
    1.897
    1.021--my answer (but got several different answers close to 1)
    1.682


    3. The table below gives data points for the continuous function y = f(x)

    http://imagizer.imageshack.us/v2/800x600q90/706/khd9.jpg

    Approximate the area under the curve y = f(x) on the interval [0, 2] using left-hand endpoints and 10 equal subdivisions. You get Area ≈


    96.8
    454----my answer
    88.8
    90.8
    444


    4. Consider the curve and the region under f (x) between x = 1 and x = 3, which is graphed below.

    http://imagizer.imageshack.us/v2/800x600q90/23/f1o5.jpg

    Suppose L is the left-hand endpoint Riemann sum with 15 subdivisions, R is the right-hand endpoint Riemann sum with 15 subdivisions, and A is the true area of this region. Which of the following is correct?


    R < L < A
    L < A < R----my answer
    L = A = R
    R < A < L
    A < R < L


    5. The function y = f(x) is graphed below:

    http://imagizer.imageshack.us/v2/800x600q90/841/a45g.jpg

    Which of the following Riemann sums yields the exact area under the curve on the interval [0, 6]?

    I. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and right-hand endpoints are used.

    II. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and midpoints are used.

    III.R=E(above=6)below=k=1 f(wk)deltaxk , where 6 equal subdivisions and right-hand endpoints are used.


    I only
    II only
    III only
    I and II only---my answer
    I, II, and III


    6. Here is a graph of the function:

    http://imagizer.imageshack.us/v2/800x600q90/826/zebj.jpg

    Estimate the total area under this curve on the interval [0, 12] with a Riemann sum using 36 equal subdivisions and circumscribed rectangles. Hint: use symmetry to make this problem easier.


    57.340
    86.634-- my answer
    14.439
    49.914
    28.044
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  2. #2
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    Re: 6 problems! Any help appreciated :)

    Let's start with #1. You divide [0,1] into 10 equal sub-intervals [0,0.1],[0.1,0.2], ... ,[0.9,1]. Then you estimate the area under the curve from 0 to 0.1 (for example) by a rectangle with width 0.1 and height f(0.05) - this is the midpoint rule (0.05 is the midpoint of [0,0.1]). So you add up the areas from all 10 sub-intervals and that's the estimate of your integral.

    If you show me your work, I could check it for you.

    Numbers 2, 3, and 6 are similar, just with different methods of finding the height of the rectangles.

    - Hollywood
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