Question

suppose
f(x)=left{\begin{array}{ll}6 - 3x & \text { if } 0 leq x<2, \\ 3x - 6 & \text { if } 2 leq x leq 4.end{array}
ight.
evaluate the definite integral by interpreting it in terms of signed area.
int_{0}^{4} f(x) d x=
suggestion: draw a picture of the region whose signed area is represented by the integral. then find the signed area using formulas from high school geometry.

Explanation:

Step1: Split the integral

Since \(f(x)\) is a piece - wise function, we split \(\int_{0}^{4}f(x)dx\) into \(\int_{0}^{2}f(x)dx+\int_{2}^{4}f(x)dx\).
For \(y = 6-3x\) (when \(0\leq x<2\)):
When \(x = 0\), \(y=6\); when \(x = 2\), \(y=0\).
For \(y = 3x - 6\) (when \(2\leq x\leq4\)):
When \(x = 2\), \(y = 0\); when \(x=4\), \(y=6\).

Step2: Calculate the area of the first triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\).
For the function \(y = 6-3x\) on the interval \([0,2]\), the base \(b_1=2\) and the height \(h_1 = 6\). So, \(A_1=\frac{1}{2}\times2\times6=6\).

Step3: Calculate the area of the second triangle

For the function \(y = 3x - 6\) on the interval \([2,4]\), the base \(b_2=2\) and the height \(h_2 = 6\). So, \(A_2=\frac{1}{2}\times2\times6=6\).

Answer:

\(12\)