Problem

Find the net signed area between the graph of the function $f(x)=-x-3$ and the $x$-axis over the interval $[-8,1]$, illustrated in the following image. Submit your answer as an exact value.

Solution

Step 1 :The net signed area between a function and the x-axis over an interval can be found by integrating the function over that interval. In this case, we need to integrate the function \(f(x)=-x-3\) over the interval \([-8,1]\).

Step 2 :The integral of a function gives the area under the curve, but since we are looking for the net signed area, areas below the x-axis will be subtracted from areas above the x-axis.

Step 3 :Since the function \(f(x)=-x-3\) is always below the x-axis over the interval \([-8,1]\), the net signed area will be negative.

Step 4 :The absolute value of the area between the function and the x-axis over the interval \([-8,1]\) is \(9/2\). However, since the function is below the x-axis over this interval, the net signed area is the negative of this value.

Step 5 :The net signed area between the graph of the function \(f(x)=-x-3\) and the x-axis over the interval \([-8,1]\) is \(-\boxed{\frac{9}{2}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/wR7xJZTjx3/

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