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Math Help - integrals pls help asap

  1. #1
    Newbie memena's Avatar
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    Unhappy integrals pls help asap

    Estimate the area under the graph of f(x) = e^-x^2 from x = -2 to x = 2 using four approximating rectangles and midpoints.

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by memena View Post
    Estimate the area under the graph of f(x) = e^-x^2 from x = -2 to x = 2 using four approximating rectangles and midpoints.

    divide the interval [-2,2] into four equal sub-intervals. that is, we partition it into the intervals: [-2,-1], [-1,0], [0,1] and [1,2]

    now, take the midpoints of these rectangles, and label them by x_i for i = 0,1,2,.... so we have x_0 = -1.5, ~x_1 = -0.5, ~x_2 = 0.5, \mbox{ and } x_3 = 1.5

    now we can use a finite Riemann sum to estimate the area. thus, an estimate of the area is given by:

    A = \sum_{i = 0}^3 f(x_i) \Delta x

    where \Delta x = x_1 - x_0 = x_2 - x_1 = ... = 1 and of course, f(x) = e^{-x^2}
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    to expound:

    since \Delta  x = 1, the area estimate is given by:

    \sum_{i = 0}^3 f(x_i) = f(x_0) + f(x_1) + f(x_2) + f(x_3)

    now i hope you can continue from there

    if you are not clear on something, say so
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  4. #4
    Newbie memena's Avatar
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    ummm

    ok do i draw this graph and then add the rectangles, if so can u tell me what it is supposed to look like, im sorry i am completely lost, i don't understand anything!
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  5. #5
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    Quote Originally Posted by memena View Post
    ok do i draw this graph Mr F says: Yes, use a graphics calculator.

    and then add the rectangles, Mr F says: Yes, add the area of each of the four rectangles. As has already been stated, the total area is f(x_0) + f(x_1) + f(x_2) + f(x_3), where x_0 = -1.5, ~x_1 = -0.5, ~x_2 = 0.5 \, and  \, x_3 = 1.5.

    if so can u tell me what it is supposed to look like, Mr F says: Draw the graph (use a graphics calculator to get it) and draw in the rectangles. Then you'll know what it looks like.

    im sorry i am completely lost, i don't understand anything! Mr F says: But you must surely have some examples to follow ....?
    ..
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by memena View Post
    ok do i draw this graph and then add the rectangles, if so can u tell me what it is supposed to look like, im sorry i am completely lost, i don't understand anything!
    yes, you can draw a diagram to visualize it.

    here:

    basically, your finding the area of the 4 rectangles we have in the diagram, that is, length times width for each. the height of each rectangle is given by f(x_i), the widths are all one
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