Question

evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles. enter the sum of all shaded areas in terms of π. provide your answer below: f(x)=√(6x - x²) g(x)=1 - |x - 7| h(x)=4 - |x - 12|

Explanation:

Step1: Analyze the semi - circle

The function $f(x)=\sqrt{6x - x^{2}}$ can be rewritten as $(x - 3)^{2}+y^{2}=9,y\geq0$, which is a semi - circle with radius $r = 3$. The area of a semi - circle is $A_{1}=\frac{1}{2}\pi r^{2}$. Substituting $r = 3$ into the formula, we get $A_{1}=\frac{1}{2}\pi\times3^{2}=\frac{9\pi}{2}$.

Step2: Analyze the small triangle

For the function $g(x)=1-|x - 7|$, when $x = 7$, $y = 1$. The base and height of the small triangle are both $1$, so its area $A_{2}=\frac{1}{2}\times1\times1=\frac{1}{2}$.

Step3: Analyze the large triangle

For the function $h(x)=4-|x - 12|$, when $x = 12$, $y = 4$. The base of the large triangle is $8$ and the height is $4$. Using the formula for the area of a triangle $A=\frac{1}{2}bh$, we have $A_{3}=\frac{1}{2}\times8\times4 = 16$.

Answer:

$\frac{9\pi}{2}+\frac{1}{2}+16$