Use the limit process to find the area of the region between the graph of the function and the x-axis over the given y-interval.

g(y) = 4y2 − y3, 1 ≤ y ≤ 3

To find the area of the region between the graph of the function and the x-axis over the given y-interval using the limit process, we need to follow these steps:

Step 1: Divide the y-interval into many small subintervals.
In this case, the given y-interval is [1, 3]. Let's choose a value for n (the number of subintervals). The larger the value of n, the more accurate the result will be. For simplicity, let's choose n = 4.

Step 2: Determine the width of each subinterval.
To find the width of each subinterval, we divide the length of the y-interval by the number of subintervals.
Width of each subinterval = (3 - 1) / 4 = 0.5

Step 3: Choose a representative value for y in each subinterval.
Since we divided the y-interval into four subintervals, we need to choose four representative values for y. Let's denote these values as y1, y2, y3, and y4, with y1 representing the first subinterval, y2 representing the second, and so on.
Using the chosen subinterval width of 0.5, we have:
y1 = 1 + (0.5/2) = 1.25
y2 = y1 + 0.5 = 1.75
y3 = y2 + 0.5 = 2.25
y4 = y3 + 0.5 = 2.75

Step 4: Evaluate the function at each representative value of y.
Evaluate the function g(y) = 4y^2 - y^3 at each of the representative values we found. This will give us the corresponding y-coordinate on the graph.
g(y1) = 4(1.25^2) - (1.25^3)
g(y2) = 4(1.75^2) - (1.75^3)
g(y3) = 4(2.25^2) - (2.25^3)
g(y4) = 4(2.75^2) - (2.75^3)

Step 5: Calculate the area of each rectangle.
The width of each rectangle is the same (0.5), so we only need to calculate the height of each rectangle by using the values we obtained in the previous step.

Height of rectangle 1 = g(y1) = 4(1.25^2) - (1.25^3)
Height of rectangle 2 = g(y2) = 4(1.75^2) - (1.75^3)
Height of rectangle 3 = g(y3) = 4(2.25^2) - (2.25^3)
Height of rectangle 4 = g(y4) = 4(2.75^2) - (2.75^3)

Step 6: Sum the areas of all the rectangles.
To find the total area, we sum the areas of all the rectangles.
Area ≈ Width × (Height of rectangle 1 + Height of rectangle 2 + Height of rectangle 3 + Height of rectangle 4)
≈ 0.5 × (Height of rectangle 1 + Height of rectangle 2 + Height of rectangle 3 + Height of rectangle 4)

By following these steps, you can use the limit process to find the area of the region between the graph of the function g(y) = 4y^2 - y^3 and the x-axis over the given y-interval [1, 3].