Question

How can I get the area of the shaded region?

Answer 1

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) = $\frac 1 2 (9)(9) = \frac{81}{2} = 40.5$

Now we have to subtract

$\displaystyle \int_0^9 (3x - x^{\frac 3 2})dx = \frac 3 2 x^2 - \frac 2 5 x^{\frac 5 2} \bigg|_0^9 = \frac 3 2 (9^2) - \frac 2 5 (9^{\frac 5 2}) = \frac{1}{2}(243) - \frac 2 5 (243) = 24.3$

Area = 40.5 - 24.3 = 16.2

Answer: 16.2