Question

Use geometry to evaluate the integral from 0 to 6 of the function f of x, dx for f of x equals 3 for x less than or equal to 3 and equals the quantity 6 minus x for x greater than 3.
27
13.5
12
10.5

Answer 1

Check the picture below.

$\bf f(x)= \begin{cases} 3&x\le 3\\ 6-x&x\ \textgreater \ 3 \end{cases}\qquad \displaystyle \int_{0}^{6}~f(x)\cdot dx$

so, you simply have a 3x3 square and a triangle with a base of 3 and a height of 3, get their areas, sum them up, it'd come out to 9 + 4.5 or 13.5.