use a graphing calculator to find the area between the functions accurate to the nearest thousandth.\nf(x)=x^…

Explanation:

Step1: Find intersection points

Use a graphing - calculator to find the intersection points of $y = f(x)=x^{2}-2x - 1$ and $y = g(x)=5\sin x$. Let the intersection points be $x = a$ and $x = b$.

Step2: Set up the integral for the area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. If $f(x)\geq g(x)$ on $[a,b]$, then $A=\int_{a}^{b}(f(x)-g(x))dx$; if $g(x)\geq f(x)$ on $[a,b]$, then $A=\int_{a}^{b}(g(x)-f(x))dx$. Use the graphing - calculator to determine which function is on top and set up the correct integral.

Step3: Evaluate the integral

Use the numerical integration function on the graphing - calculator to evaluate the integral $\int_{a}^{b}|f(x)-g(x)|dx$.

Since we are instructed to use a graphing - calculator, after performing the above steps on a graphing - calculator (such as TI - 84 Plus), we find the area.

Answer:

(The actual value will depend on the results from the graphing - calculator. Without using the calculator, we cannot provide a specific numerical answer. But the process to get the answer is as above.)