Question

question 4
evaluate the integral below by using what we know about the areas in the figure.
the areas of the labeled regions are
a1 = 6
a2 = 3
a3 = 1
a4 = 2
\\(\int_{0}^{7} f(x) dx = \square\\)
enter your answer as an integer
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Explanation:

Step1: Split integral by intervals

$$\int_{0}^{7} f(x) dx = \int_{0}^{3} f(x) dx + \int_{3}^{5} f(x) dx + \int_{5}^{7} f(x) dx$$

Step2: Map to region areas

Areas above x-axis are positive, below are negative:
$\int_{0}^{3} f(x) dx = A1 = 6$, $\int_{3}^{5} f(x) dx = -A2 = -3$, $\int_{5}^{7} f(x) dx = A3 = 1$

Step3: Calculate sum

$$6 + (-3) + 1 = 4$$

Answer:

$4$