Question

exercises 1.2 the definite integral
score: 80/170 answered: 8/17
question 9
evaluate the integral below by interpreting
it in terms of areas in the figure.
the areas of the labeled regions are
a1= 8, a2=4, a3=2 and a4=2
v = \\(\int_{0}^{7} (1 + f(x)) dx\\)
enter your answer as a whole number
question help: video 1 video 2 message instructor
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Explanation:

Step1: Split the integral

$$V = \int_{0}^{7} 1\ dx + \int_{0}^{7} f(x)\ dx$$

Step2: Evaluate $\int_{0}^{7} 1\ dx$

This is the area of a rectangle with width $7-0=7$ and height $1$, so:
$$\int_{0}^{7} 1\ dx = 7 \times 1 = 7$$

Step3: Evaluate $\int_{0}^{7} f(x)\ dx$

Regions above the x-axis add positive area, below add negative area:
$$\int_{0}^{7} f(x)\ dx = A1 - A2 + A3 = 8 - 4 + 2 = 6$$

Step4: Sum the two results

$$V = 7 + 6$$

Answer:

13