Use the limit process to find the area of the region between the graph of the function and the y-axis over the given y-interval.
f(y) = 7y, 0 ≤ y ≤ 2
The y-interval given says that the y-value would greater than or equal to 0 as well as less than or equal to 2. This means that you can calculate the limit, or in this case the area, by inserting 2 since it is the maximum number set by the interval and thus the limit for the area. When you multiply it you get f(x) = 14, which is your answer. As for the graph, this is how it should look like. Since the limit of the interval is 2, the y-axis of the region should be from 0 to 2. At 2, place a horizontal dashed line. The dashed line should end when it reaches the x-value equal to the area found, in your case, 14. Finally, draw a solid diagonal line that meets with the end of the horizontal line, creating a triangle. Now you've drawn the region on the graph! The end :)
To find the area of the region between the graph of the function f(y) = 7y and the y-axis over the given y-interval [0, 2], we can use the limit process.
The area can be approximated by dividing the y-interval into n subintervals, and finding the area of each subinterval. We can then take the limit as n approaches infinity to find the exact area.
Step 1: Divide the y-interval into n equal subintervals.
In this case, the y-interval is [0, 2]. Let's choose a positive integer n to divide this interval into n subintervals. Each subinterval will then have length Δy = (2-0)/n = 2/n.
Step 2: Choose a representative point within each subinterval.
To approximate the area of each subinterval, we need to choose a representative point within each subinterval. Let's denote these points as y_i, where i ranges from 1 to n. The total number of representative points will be n.
Step 3: Find the area of each subinterval.
For each subinterval, we can approximate the area as the product of the function value at the representative point and the length of the subinterval. So, the area of each subinterval can be given by A_i = f(y_i) * Δy.
In this case, f(y) = 7y, so the area of each subinterval can be written as A_i = 7y_i * (2/n).
Step 4: Sum up the areas of all subintervals.
To find the total area, we need to sum up the areas of all the subintervals. This can be done by adding up all the individual areas from i = 1 to n: A_total = Σ A_i = Σ [7y_i * (2/n)].
Step 5: Take the limit as n approaches infinity.
To find the exact area, we need to take the limit as n goes to infinity of the sum of the individual areas: A_exact = lim(n→∞) Σ [7y_i * (2/n)].
By taking the limit as n approaches infinity, we can get an accurate approximation of the area between the graph of the function f(y) = 7y and the y-axis over the y-interval [0, 2].