Question

homework 1: 1.1-1.4 review
score: 4/17 answered: 1/13
question 2
evaluate the integral below by interpreting
it in terms of areas in the figure.
the areas of the labeled regions are
a1= 5, a2=3, a3=2 and a4=1
v = \\(\int_{0}^{7} f(x) dx\\)
v =
question help: video
(figure: y=f(x) with regions a1 (yellow, 0 - 3), a2 (cyan, 3 - 5), a3 (yellow, 5 - 7), a4 (cyan, 7 - 10); areas a1=5, a2=3, a3=2, a4=1; figure is not to scale)

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: Assign area values

Areas above x-axis are positive, below are negative:
$\int_{0}^{3} f(x) dx = 5$, $\int_{3}^{5} f(x) dx = -3$, $\int_{5}^{7} f(x) dx = 2$

Step3: Calculate total value

$$5 + (-3) + 2 = 4$$

Answer:

4