what is the area of the region between the graphs of f(x)=2x² + 5x and g(x)= -x² - 6x + 4 from x = -4 to x =…

Explanation:

Step1: Find the difference function

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. First, find $f(x)-g(x)$: [ \begin{align*} f(x)-g(x)&=(2x^{2}+5x)-(-x^{2}-6x + 4)\ &=2x^{2}+5x + x^{2}+6x - 4\ &=3x^{2}+11x - 4 \end{align*} ]

Step2: Calculate the definite - integral

Now, calculate $\int_{-4}^{0}(3x^{2}+11x - 4)dx$. Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have: [ \begin{align*} \int_{-4}^{0}(3x^{2}+11x - 4)dx&=\left[x^{3}+\frac{11}{2}x^{2}-4x\right]_{-4}^0\ &=(0^{3}+\frac{11}{2}\times0^{2}-4\times0)-((-4)^{3}+\frac{11}{2}\times(-4)^{2}-4\times(-4))\ &=0-\left(-64+\frac{11}{2}\times16 + 16\right)\ &=-\left(-64 + 88+16\right)\ &=-\left(40\right)\ & = 40 \end{align*} ]

Answer:

B. 40