question the piece - wise function f(x) is gr...
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:
$\frac{9\pi}{2}+6$