question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following…

question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫-7,5f(x)dx enter an exact answer.

question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫-7,5f(x)dx enter an exact answer.

Answer

Explanation:

Step1: Split the integral

The integral $\int_{-7}^{5}f(x)dx=\int_{-7}^{-1}f(x)dx+\int_{-1}^{5}f(x)dx$.

Step2: Evaluate $\int_{-7}^{-1}f(x)dx$

The graph from $x = - 7$ to $x=-1$ is a semi - circle. The radius $r = 3$. The area of a semi - circle is $A=\frac{1}{2}\pi r^{2}$. So $\int_{-7}^{-1}f(x)dx=\frac{1}{2}\pi(3)^{2}=\frac{9\pi}{2}$.

Step3: Evaluate $\int_{-1}^{5}f(x)dx$

The graph from $x=-1$ to $x = 5$ is a triangle. The base $b=6$ and the height $h = 2$. The area of a triangle is $A=\frac{1}{2}bh$. So $\int_{-1}^{5}f(x)dx=\frac{1}{2}(6)(2)=6$.

Step4: Combine the results

$\int_{-7}^{5}f(x)dx=\frac{9\pi}{2}+6$.

Answer:

$\frac{9\pi}{2}+6$