Question

Given the function f(x) = 3x + 10, find the net signed area between f(x) and the x-axis over the interval [−6, −2]. Do not include any units in your answer.

Answer 1

The net signed area between f(x) = 3x + 10 and the x-axis over the interval [-6, -2] is 8.

Step-by-step explanation:

The net signed area between a function and the x-axis over an interval is found by evaluating the definite integral of the absolute value of the function within that interval. In this case, the function is f(x) = 3x + 10 and the interval is [-6, -2]. To find the net signed area, we integrate the absolute value of the function over the interval [-6, -2].

|f(x)| = |3x + 10|

Integrating |f(x)| over [-6, -2]:

∫[−6,−2] |3x + 10| dx

Since the function f(x) is a linear function, we can break up the interval into two parts:

∫[−6,−2] (3x + 10) dx = ∫[−6,−2] (3x) dx + ∫[−6,−2] (10) dx

Integrating each part:

[3x^2/2]−6,−2 + [10x]−6,−2

Substituting the values, we get:

([((3(-2)^2)/2)−((3(-6)^2)/2)] + [(10(-2))−(10(-6))]) = 8

Therefore, the net signed area between f(x) = 3x + 10 and the x-axis over the interval [-6, -2] is 8.