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

# 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 the upper - half of a circle with center $(4,0)$ and radius $r = 3$. ## Step2: Determine the area under the curve for the given integral The definite integral $\int_{1}^{4}\sqrt{-(x - 4)^2 + 9}dx$ represents the area under the curve of the upper - half of the circle from $x = 1$ to $x = 4$. The region is a quarter - circle. ## Step3: Use the area formula for a circle The area formula for a circle is $A=\pi r^{2}$. For a quarter - circle, the area $A_{quarter}=\frac{1}{4}\pi r^{2}$. Since $r = 3$, we have $A_{quarter}=\frac{1}{4}\pi\times3^{2}=\frac{9\pi}{4}$. # Answer: $\frac{9\pi}{4}$