Math Help - Integration math solution

1. Integration math solution

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

f(x) = 0.5x + 2; g(x) = 2x - 4

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