Find the area of the region bounded by the graphs of the algebraic functions.

Question

Find the area of the region bounded by the graphs of the algebraic functions. ​
​   ​

Quizplus · Accepted Answer

First, we need to find the intersection of the two functions:
$$ 2x - 3 = -3x^2 + 4x $$
Solving for x gives $ x = 1 $ and $ x = -\frac{1}{3} $. 
We can see that the function $ y = 2x - 3 $ is on top of $ y = -3x^2 + 4x $ on the interval $ [-\frac{1}{3}, 1] $. Therefore, we can find the area of the region using the definite integral:
$$ \int_{-\frac{1}{3}}^{1} (2x - 3) - (-3x^2 + 4x) \, dx = \int_{-\frac{1}{3}}^{1} 3x^2 - 2x - 3 \, dx = \left[\frac{1}{3}x^3 - x^2 - 3x \right]_{-\frac{1}{3}}^{1} = \frac{38}{9} $$
So the answer is C.

Find the Area of the Region Bounded by the Graphs