Please help me on calculus - integral

Question

Use geometry to evaluate  for 

Options:

1. 12.5
2. 25
3. 37.5
4. Cannot be found

Answer 1

Answer 2

If you try to plot f(x), you see it has two different definition before and after the domain x=5.

If we limit the domain of x with the integral limits, then we have two functions;

the f₁(x) = 5 if 0 <= x <= 5 and

f₂ (x)=10-x if 5 < x <=10.

Now the int f(x)= int ( f1 + f2 ), so integrate each function by itself and then add them up.

int f₁(x)=5x₀⁵= 5(5-0)=25

int f₂ (x)=10x-x²/2 (from 5 to 10)= 10(10)-100/2 -10(5)+25/2= 100-50-50+12.5=12.5

then int f= 25 + 12.5 =37.5