Transcribed Image Text: The areas of the regions bounded by the graph of the function f and the x-axis are labeled in the figure below. Let the function g be defined by the equation g(x) = | f(t)dt. What is the value g(-5)? Graph of f 8. 6. 14 10 9. 4 -10
To determine the value of g(-5), we first need to understand the equation that defines the function g. The equation is g(x) = ∫ f(t) dt, where ∫ represents the integral sign and f(t) is the function that we want to find the integral of.
In this case, the function f is given as a graph, and we can see that it is a piecewise function with different regions bounded by the graph and the x-axis. The areas of these regions are labeled in the figure provided.
To find the value of g(-5), we need to evaluate the integral of f(t) from the lower limit to the upper limit, where the lower limit is -5 and the upper limit is t. Since we are looking for the value of g(-5), the upper limit will be -5.
Let’s break down the integral calculation step by step:
∫ f(t) dt = ∫ f(t) dt evaluated from -5 to t
First, let’s look at the regions of the graph of f to determine the function f in each region:
Region 1: The graph of f is below the x-axis. Since the integral represents the area between the graph and the x-axis, the value of the integral in this region will be negative.
Region 2: The graph of f is above the x-axis and is a straight line. In this region, the integral will represent the area of a triangle.
Region 3: The graph of f is again below the x-axis. Similar to region 1, the integral in this region will be negative.
To calculate the integral, we need to consider the individual regions and their corresponding equations. Let’s denote the integral in each region as I1, I2, and I3.
For Region 1, the integral will be:
I1 = ∫ f(t) dt from -5 to t = -5 * (area of triangle)
Since the graph of f in this region is below the x-axis, the area of the triangle will be negative. We can calculate the area of the triangle by multiplying the length of the base (5) by the height of the triangle (4), and then dividing by 2:
I1 = -5 * (5 * 4) / 2 = -10 * 4 = -40
Therefore, the value of I1 is -40.
For Region 2, the integral will be:
I2 = ∫ f(t) dt from -5 to t = (area of triangle)
Since the graph of f in this region is above the x-axis, the area of the triangle will be positive. Using the same formula as before, we can calculate the area of the triangle:
I2 = (5 * 4) / 2 = 10
Therefore, the value of I2 is 10.
For Region 3, the integral will be:
I3 = ∫ f(t) dt from -5 to t = -5 * (area of triangle)
Similar to Region 1, the graph of f in this region is below the x-axis. We can calculate the area of the triangle and multiply it by -5:
I3 = -5 * (5 * 2) / 2 = -5 * 5 = -25
Therefore, the value of I3 is -25.
Now, let’s sum up the individual integrals to calculate g(-5):
g(-5) = I1 + I2 + I3 = -40 + 10 + (-25) = -55
Therefore, the value of g(-5) is -55.
In conclusion, by evaluating the integral of the given function f(t) from -5 to -5, we determined that the value of g(-5) is equal to -55.