current objective find the area of a region bounded by two functions that cross question calculate the area…

Explanation:

Step1: Find the difference of the functions

Let $h(x)=g(x)-f(x)=(-4x^{3}-4x^{2}+2x - 8)-(-5x^{3}+3x^{2}-8x - 8)=x^{3}-7x^{2}+10x$.

Step2: Use the definite - integral formula for area

The area $A$ between two functions $y = f(x)$ and $y = g(x)$ on the interval $[a,b]$ is $A=\int_{a}^{b}|g(x)-f(x)|dx$. Here, on $[3,7]$, we calculate $\int_{3}^{7}(x^{3}-7x^{2}+10x)dx$.

Step3: Integrate term - by - term

$\int(x^{3}-7x^{2}+10x)dx=\frac{x^{4}}{4}-\frac{7x^{3}}{3}+5x^{2}+C$.

Step4: Evaluate the definite integral

$\left[\frac{x^{4}}{4}-\frac{7x^{3}}{3}+5x^{2}\right]_{3}^{7}=\left(\frac{7^{4}}{4}-\frac{7\times7^{3}}{3}+5\times7^{2}\right)-\left(\frac{3^{4}}{4}-\frac{7\times3^{3}}{3}+5\times3^{2}\right)$. First, calculate $\frac{7^{4}}{4}-\frac{7\times7^{3}}{3}+5\times7^{2}=\frac{2401}{4}-\frac{2401}{3}+245=\frac{7203 - 9604+2940}{12}=\frac{539}{12}$. Second, calculate $\frac{3^{4}}{4}-\frac{7\times3^{3}}{3}+5\times3^{2}=\frac{81}{4}-63 + 45=\frac{81-252 + 180}{4}=\frac{9}{4}$. Then, $\frac{539}{12}-\frac{9}{4}=\frac{539 - 27}{12}=\frac{512}{12}=\frac{128}{3}$.

Answer:

$\frac{128}{3}$