the piecewise function f(x) is graphed below. use geometric formulas to evaluate the following definite…

Explanation:

Step1: Split the integral

The definite - integral $\int_{1}^{5}f(x)dx=\int_{1}^{3}f(x)dx+\int_{3}^{5}f(x)dx$.

Step2: Evaluate $\int_{1}^{3}f(x)dx$

The region under the curve $y = f(x)$ from $x = 1$ to $x = 3$ is a semi - circle with radius $r = 1$. The area of a semi - circle is $A=\frac{1}{2}\pi r^{2}$. Here, $A_1=\frac{1}{2}\pi(1)^{2}=\frac{\pi}{2}$. Since the semi - circle is above the $x$ - axis, $\int_{1}^{3}f(x)dx=\frac{\pi}{2}$.

Step3: Evaluate $\int_{3}^{5}f(x)dx$

The region under the curve $y = f(x)$ from $x = 3$ to $x = 5$ is a triangle with base $b = 2$ and height $h = 3$. The area of a triangle is $A=\frac{1}{2}bh$. Here, $A_2=\frac{1}{2}(2)(3)=3$. Since the triangle is above the $x$ - axis, $\int_{3}^{5}f(x)dx = 3$.

Step4: Combine the results

$\int_{1}^{5}f(x)dx=\int_{1}^{3}f(x)dx+\int_{3}^{5}f(x)dx=\frac{\pi}{2}+3$.

Answer:

$\frac{\pi}{2}+3$