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) = (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)) = (1)/(2)(243) - \frac 2 5 (243) = 24.3`

Area = 40.5 - 24.3 = 16.2

Answer: 16.2