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3.1 homework - area and definite integrals and areas
3.1 homework - area and definite integrals and areas
score: 40/100 attempt: 5/10
question 5
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evaluate the integral below by interpreting
it in terms of areas in the figure.
the areas of the labeled regions are
a1=7, a2=4, a3=1 and a4=2
v = \\(\int_{0}^{7} f(x) dx\\)
v =
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Explanation:

Step1: Recall Definite Integral as Area

The definite integral $\int_{a}^{b} f(x) dx$ represents the net area between the curve $y = f(x)$, the $x$-axis, from $x = a$ to $x = b$. Areas above the $x$-axis are positive, and areas below are negative.

Step2: Analyze the Interval $[0, 7]$

We need to find $\int_{0}^{7} f(x) dx$. The interval $[0, 7]$ includes regions $A1$ (from $0$ to $3$), $A2$ (from $3$ to $5$), and $A3$ (from $5$ to $7$).

• $A1 = 7$ (above $x$-axis, positive)
• $A2 = 4$ (below $x$-axis, negative)
• $A3 = 1$ (above $x$-axis, positive)

Step3: Calculate the Integral

Sum the areas with appropriate signs:
$$\int_{0}^{7} f(x) dx = A1 - A2 + A3$$
Substitute the values:
$$\int_{0}^{7} f(x) dx = 7 - 4 + 1$$
$$= 3 + 1$$
$$= 4$$

Answer:

$\boxed{4}$