Question

the function ( f ) is shown below. what is the value of ( int_{-2}^{5} f(x) , dx )? write your answer in simplest form.

Explanation:

Step1: Split integral into intervals

$\int_{-3}^{5} f(x) dx = \int_{-3}^{-1} f(x) dx + \int_{-1}^{3} f(x) dx + \int_{3}^{5} f(x) dx$

Step2: Calculate $\int_{-3}^{-1} f(x) dx$

This is area of triangle above x-axis:
Base $= -1 - (-3) = 2$, Height $= 3$
Area $= \frac{1}{2} \times 2 \times 3 = 3$

Step3: Calculate $\int_{-1}^{3} f(x) dx$

This is area of triangle above x-axis:
Base $= 3 - (-1) = 4$, Height $= 5$
Area $= \frac{1}{2} \times 4 \times 5 = 10$

Step4: Calculate $\int_{3}^{5} f(x) dx$

This is area of triangle below x-axis (negative value):
Base $= 5 - 3 = 2$, Height $= 5$
Area $= -\frac{1}{2} \times 2 \times 5 = -5$

Step5: Sum the three results

$3 + 10 + (-5) = 8$

Answer:

8