Question

use the formulas for areas of triangles and circles to evaluate the integrals of the functions graphed in the figure. enter the sum of all shaded areas in terms of π. provide your answer below. question
f(x)=\sqrt{4x - x^{2}}
g(x)=4 - |x - 8|
h(x)=\sqrt{-216 + 30x - x^{2}}
feedback
more instruction
submit

Explanation:

Step1: Analyze the shapes

The shaded region consists of two semi - circles and a triangle.

Step2: Calculate the area of the semi - circles

The first semi - circle \(f(x)=\sqrt{4x - x^{2}}\) has the equation \((x - 2)^{2}+y^{2}=4\), radius \(r_1 = 2\), and area of a semi - circle \(A_{s1}=\frac{1}{2}\pi r_1^{2}=\frac{1}{2}\pi(2)^{2}=2\pi\). The third semi - circle \(h(x)=\sqrt{- 216 + 30x - x^{2}}\) has the equation \((x - 15)^{2}+y^{2}=9\), radius \(r_2 = 3\), and area of a semi - circle \(A_{s2}=\frac{1}{2}\pi r_2^{2}=\frac{9}{2}\pi\)

Step3: Calculate the area of the triangle

The triangle \(g(x)=4|x - 8|\) has base \(b = 4\) and height \(h=6\). Area of a triangle \(A_{t}=\frac{1}{2}\times b\times h=\frac{1}{2}\times4\times6 = 12\)

Step4: Calculate the total shaded area

The total shaded area \(A=A_{s1}+A_{t}+A_{s2}=2\pi+12+\frac{9}{2}\pi=\frac{4\pi + 24+9\pi}{2}=\frac{13\pi + 24}{2}\)

Answer:

\(\frac{13\pi + 24}{2}\)