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Math Help - Areas of Common Geometric Figures homework help

  1. #1
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    Areas of Common Geometric Figures homework help

    wouldn't letters d, e, and f be the same answer??? since your taking the area over the same interval???
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  2. #2
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    Re: Areas of Common Geometric Figures homework help

    No, they would be different since the integrand is different in each case.
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    Re: Areas of Common Geometric Figures homework help

    Hello, asilvester635!

    Wouldn't letters d, e, and f be the same answer?
    . . Certainly not!

    Did you realize that some answers are negative?
    Didn't you make any sketches?


    (e)\;\int^6_{\text{-}4} |f(x)|\,dx

    All the regions below the x-axis are reflected upward!
    Code:
                               2|
                            ..* * *..               *
                          *:::::|:::::*           *:::*
          *..           *:::::::|:::::::*       *:::::::*
          ::::*..      *::::::::|::::::::*    *:::::::::::*
          ::::::::*.  ::::::::::|:::::::::: *:::::::::::::::*
      - - + - - - - - * - - - - * - - - - * - - - - + - - - - * - -
         -4          -2         |         2         4         6
                                |
    We want the area of the shaded region.



    (f)\;\int^6_{\text{-}4}[f(x)+2]\,dx

    The entire graph is moved 2 units upward!
    Code:
                                |
                               4+                   *
                                |                 *:::*
                                |               *:::::::*
                                |             *:::::::::::*
                                |           *:::::::::::::::*
                     .*        2+         *:::::::::::::::::::*
                 .*:::|         |         |:::::::::::::::::::|
             .*:::::::|*        |        *|:::::::::::::::::::|
          *:::::::::::|:*.      |      .*:|:::::::::::::::::::|
          |:::::::::::|:::*..   |   ..*:::|:::::::::::::::::::|
      - - + - - - - - + - - - * * * - - - + - - - - + - - - - + - -
         -4          -2         |         2         4         6
                                |
    We want the area under the graph.

    Got it?
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    Re: Areas of Common Geometric Figures homework help

    holy crap... yes i got it.... do we minus the area if it's below the x-axis??? like for the small triangle's area under the x-axis is 1 and he semicircle's area is 2pi and that big triangle above the x-axis area is 4....
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    Re: Areas of Common Geometric Figures homework help

    Hello, asilvester635!

    Do we subtract the area if it's below the x-axis?

    Yes . . . and that is the "shortcut" way of looking at it.
    Actually, since you are working with areas,
    . . you should have been aware of this long ago.


    Consider the area bounded by: y \:=\:2x-4 and the coordinate axes.

    Code:
            |
            |      /
        ----+-----*----
            |::::/2
            |:::/
            |::/
            |:/
            |/
            *
           /|
            |
    The integral is: . A \;=\;\int^2_0(2x-4)\,dx

    We have: . x^2 - 4x\,\bigg|^2_0 \;=\;(4-8) - (0-0) \;=\;{\color{red}-}4 . (negative four)


    If the region is below the x-axis, the value of the integral is negative.
    . . (You can call it "negative area" if you like.)
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  6. #6
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    Re: Areas of Common Geometric Figures homework help

    Quote Originally Posted by asilvester635 View Post
    ... do we minus the area if it's below the x-axis???
    only if the lower limit of integration < upper limit of integration
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