Final answer:
The net signed area between the function f(x) = -2x and the x-axis over the interval [-4,9] is calculated by summing the negative area from -4 to 0 and the negative area from 0 to 9, resulting in a net area of -65.
Step-by-step explanation:
To find the net signed area between the function f(x) = -2x and the x-axis over the interval [−4,9], we can use the concept of definite integrals. The area above the x-axis is considered positive, while the area below the x-axis is considered negative. The function f(x) = -2x is a straight line, which means we can break the integral into two parts: one for the negative interval (from -4 to 0) and one for the positive interval (from 0 to 9).
First, calculate the area from -4 to 0:
- Integral of f(x) from -4 to 0 = Integral of -2x from -4 to 0 = (-2)(x^2)/2 from -4 to 0 = 0 - ( -2(-4)^2/2 ) = 16.
This area is negative since it is below the x-axis.
Next, calculate the area from 0 to 9:
- Integral of f(x) from 0 to 9 = Integral of -2x from 0 to 9 = (-2)(x^2)/2 from 0 to 9 = (-2)(9^2)/2 - 0 = -81.
This area is also negative and it is the larger portion under the x-axis.
The net signed area is the sum of these two areas:
Net signed area = 16 + (-81)
= -65.