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Math Help - Area under the graph using Right and Left end-points

  1. #1
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    Area under the graph using Right and Left end-points


    Problem 2:
    Estimate the area under the graph of f(x) = 4 cos(x) from x = 0 to x = π/2. (Round the answer to four decimal places.)(a) Use four approximating rectangles and right endpoints.

    My work and answer (which is wrong according to the site)
    Delta x= Pi/2 / 4 = pi/8
    R4= pi/8 (3.8+3.4+2.5+1.6)
    R4= 4.4375


    (b) b) Use four approximating rectangles and left endpoints.
    L4= pi/8 f(pi/8) + pi/8 f(pi/4) +pi/8 f(3pi/4)
    L4= pi/8 (4 + 3.9 +3.3 + 2.7)
    L4=5.4585


    I made a graph so these values are my estimates

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    Re: Area under the graph using Right and Left end-points

    Quote Originally Posted by Steelers72 View Post

    Problem 2:
    Estimate the area under the graph of f(x) = 4 cos(x) from x = 0 to x = π/2. (Round the answer to four decimal places.)(a) Use four approximating rectangles and right endpoints.

    My work and answer (which is wrong according to the site)
    Delta x= Pi/2 / 4 = pi/8
    R4= pi/8 (3.8+3.4+2.5+1.6)
    R4= 4.4375


    (b) b) Use four approximating rectangles and left endpoints.
    L4= pi/8 f(pi/8) + pi/8 f(pi/4) +pi/8 f(3pi/4)
    L4= pi/8 (4 + 3.9 +3.3 + 2.7)
    L4=5.4585


    I made a graph so these values are my estimates

    I only checked the value I bolded, and it is wrong.
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  3. #3
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    Re: Area under the graph using Right and Left end-points

    Your \Delta x is correct.

    Remember, part (a) instructs you to use right end points.

    As such,
    f(x)\approx \Delta x \left(f(\tfrac{\pi}{8})+f(\tfrac{\pi}{4})\right +f(\tfrac{3\pi}{8})+f(\tfrac{\pi}{2}))
    is the right Riemann approximation with four (evenly spaced) rectangles.

    EDIT: GO RAVENS!!
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    Re: Area under the graph using Right and Left end-points

    Thanks for your replies. So which values do I multiply the values you gave me by? so it's Pi/8 ( f(pi/8)+f(pi/4)+...)and so on. I'm lost on how to find the values needed (as you saw from 3.8 you bolded. Do I make graphs? Really confused on that.

    6. Enough said
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    Re: Area under the graph using Right and Left end-points

    Quote Originally Posted by Steelers72 View Post
    Thanks for your replies. So which values do I multiply the values you gave me by? so it's Pi/8 ( f(pi/8)+f(pi/4)+...)and so on. I'm lost on how to find the values needed (as you saw from 3.8 you bolded. Do I make graphs? Really confused on that.
    f(x)=4\cos(x)

    f(\tfrac{\pi}{8})=4\cos(\tfrac{\pi}{8})\approx 3.69551813004515

    f(\tfrac{\pi}{4})=4\cos(\tfrac{\pi}{4})\approx 2.82842712474619

    f(\tfrac{3\pi}{8})=4\cos(\tfrac{3\pi}{8})\approx 1.53073372946037

    f(\tfrac{\pi}{2})=4\cos(\tfrac{\pi}{2}) = 0

    Right Riemann Approx: f(x)\approx \Delta x \left(f(\tfrac{\pi}{8})+f(\tfrac{\pi}{4})\right +f(\tfrac{3\pi}{8})+f(\tfrac{\pi}{2}))
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    Re: Area under the graph using Right and Left end-points

    Thank you! I got 3.1631 as the R4 value.

    From the left, how would the values change?
    It would still be pi/8 (f(pi/8) + f (pi/4) + f(p(3pi/8) + f(pi/2)) and I know from the left side sum it is an over-estimation as compared to the under-estimation from the right.

    thanks again for all of your help.
    Last edited by Steelers72; February 9th 2013 at 07:45 PM.
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  7. #7
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    Re: Area under the graph using Right and Left end-points

    Quote Originally Posted by Steelers72 View Post
    Thank you! I got 3.1631 as the R4 value.

    From the left, how would the values change?
    It would still be pi/8 (f(pi/8) + f (pi/4) + f(p(3pi/8) + f(pi/2)) and I know from the left side sum it is an over-estimation as compared to the under-estimation from the right.

    thanks again for all of your help.
    A LEFT Riemann sum uses LEFT endpoints.

    Left Riemann Approximation: f(x)\approx \Delta x \left(f(0)+ f(\tfrac{\pi}{8})+f(\tfrac{\pi}{4}) +f(\tfrac{3\pi}{8})\right)

    Literally draw the rectangles.
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    Re: Area under the graph using Right and Left end-points

    Will do. Thanks again!
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