Question

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Answer 1

$$f(x)=x-2, 2\leq x\leq 4$$ $$g(x)=\sqrt{4-(x-6)^2}+2, 4\leq x\leq 8$$ $$h(x)=x-6, 8\leq x\leq 10$$

$f(x)=x-2, 2\leq x\leq 4$

$$l_1=(4-2)\sqrt{2}=2\sqrt{2}$$

$g(x)=\sqrt{4-(x-6)^2}+2, 4\leq x\leq 8$

Hemicircle $(x-6)^2+(y-2)^2=4, 2\leq y\leq4$

$$l_2=\dfrac{1}{2}(2\pi(2)) =2\pi$$

$h(x)=x-6, 8\leq x\leq 10$

$$l_3=(10-8)\sqrt{2}=2\sqrt{2}$$ $$l=l_1+l_2+l_3$$ $$=2\sqrt{2}+2\pi +2\sqrt{2}$$ $$=4\sqrt{2}+2\pi \ (units)$$ $$l=4\sqrt{2}+2\pi \ (units)$$