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Math Help - Integration math solution

  1. #1
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    Integration math solution

    Find the first-quadrant area bounded by the axes and the functions

    f(x) = 0.5x + 2; g(x) = 2x - 4
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  2. #2
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    Hello, urmiprincess!

    Did you make a sketch?


    Find the first-quadrant area bounded by the axes and the functions:

    . . \begin{array}{cccc}f(x) &= & \frac{1}{2}x +2 & {\color{blue}[1]}\\ g(x) &=& 2x  - 4 & {\color{blue}[2]}\end{array}
    Code:
              |                       B
              |                    ...o (4,4)
              |              ...*:::* :
              |        ...*:::::::*   :
              |  ...*:::::::::::*     :
            A o2::::::::::::::*       :
              |:::::::::::::*         :
              |:::::::::D:*           : E
          - - o - - - - o - - - - - - + -
             O|       * 2             4 
              |     *
              |   *
              | *
            C *-4
              |

    \text{Line }{\color{blue}[1]} \:=\:AB
    \text{Line }{\color{blue}[2]} \:=\:CB
    . . They intersect at B(4,4)

    We want the area of quadrilateral ABDO
    . . Drop perpendicular BE

    Trapezoid ABEO has: . h = OE = 4,\;b_1 = AO = 2,\;b_2 = BE = 4
    . . Its area is: . \frac{4}{2}(2 + 4) \:=\:12

    Right triangle BED has: . b = DE = 2,\;h = BE = 4
    . . Its area is: . \frac{1}{2}(2)(4) \:=\:4


    Therefore: . \text{Area}(ABDO) \;=\;12 - 4 \:=\:\boxed{8}

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