the function f(x) = √(-(x - 4)² + 9) is graph...

# Explanation: ## Step1: Identify the geometric shape The function $y = \sqrt{-(x - 4)^2+9}$ can be rewritten as $(x - 4)^2+y^2=9$ ($y\geq0$), which represents a semi - circle with center $(4,0)$ and radius $r = 3$. ## Step2: Determine the area of the semi - circle The definite integral $\int_{1}^{4}\sqrt{-(x - 4)^2 + 9}dx$ represents the area under the curve of the semi - circle from $x = 1$ to $x=4$. The area of a full circle is $A=\pi r^{2}$, and for a semi - circle $A_{semicircle}=\frac{1}{2}\pi r^{2}$. Here, the region from $x = 1$ to $x = 4$ is a quarter - circle. ## Step3: Calculate the area The area of a quarter - circle with radius $r = 3$ is $A=\frac{1}{4}\pi r^{2}$. Substitute $r = 3$ into the formula: $A=\frac{1}{4}\pi\times3^{2}=\frac{9\pi}{4}$. # Answer: $\frac{9\pi}{4}$