let r be the region bounded by the functions f(x)=-4x² + 10x and g(x)=-5 as shown in the diagram below. find…

Explanation:

Step1: Find intersection points

Set $-4x^{2}+10x=-5$. Rearrange to $4x^{2}-10x - 5=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 4$, $b=-10$, $c=-5$, we have $x=\frac{10\pm\sqrt{(-10)^{2}-4\times4\times(-5)}}{2\times4}=\frac{10\pm\sqrt{100 + 80}}{8}=\frac{10\pm\sqrt{180}}{8}=\frac{10\pm6\sqrt{5}}{8}=\frac{5\pm3\sqrt{5}}{4}$.

Step2: Set up integral for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ is $A=\int_{a}^{b}[f(x)-g(x)]dx$, where $a$ and $b$ are the intersection - points. Here, $A=\int_{\frac{5 - 3\sqrt{5}}{4}}^{\frac{5 + 3\sqrt{5}}{4}}(-4x^{2}+10x+5)dx$.

Step3: Integrate the function

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

Step4: Evaluate the definite - integral

$A=\left[-\frac{4}{3}x^{3}+5x^{2}+5x\right]_{\frac{5 - 3\sqrt{5}}{4}}^{\frac{5 + 3\sqrt{5}}{4}}$. Using a calculator: $A\approx28.473$.

Answer:

$28.473$