Let's work on the triangle first. The x intercept of 2y=27-3x is
0 = 27 - 3x
3x = 27
x = 9
So A(9,0)
B is the meet of 2y=27-3x and y=3x
2y=27-y
3y=27
y=9
y is the height of OAB; we don't really need x
Area(OAB) = [tex]\frac 1 2 (9)(9) = \frac{81}{2} = 40.5[/tex]
Now we have to subtract
[tex]\displaystyle \int_0^9 (3x - x^{\frac 3 2})dx = \frac 3 2 x^2 - \frac 2 5 x^{\frac 5 2} \bigg|_0^9 = \frac 3 2 (9^2) - \frac 2 5 (9^{\frac 5 2}) = \frac{1}{2}(243) - \frac 2 5 (243) = 24.3[/tex]
Area = 40.5 - 24.3 = 16.2
Answer: 16.2
