PLEASE HELP WILL GIVE 50 POINTS. In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered in this unit. Sketch the graph of the function y = 20x − x2, and approximate the area under the curve in the interval [0, 20] by dividing the area into the given numbers of rectangles.
1. Use five rectangles to approximate the area under the curve.
2. Use 10 rectangles to approximate the area under the curve.
3. Calculate the area under the curve using rectangles as their number becomes arbitrarily large (tends to infinity). You do not need to sketch the rectangles.

Step-by-step explanation:

Part A. Using five rectangles.

I will use the left end-point approximation.

1. Divide the interval [0,20] into five equal subintervals

Δx = (20 - 0)/5 = 4

The intervals are:

[0,4], [4,8], [8,12], [12,16], [16,20]

2. Evaluate the function at the left end-points and calculate the areas

Fig. 1 shows the graph with five rectangles.

Part B. Using 10 rectangles

1. Divide the interval [0,20] into 10 equal subintervals

Δx = (20 - 0)/10 = 2

The intervals are:

[0,2], [2,4], [4,6], [6,8], [8,10], [10,12], [12,14], [14,16], [16,18], [18,20]

2. Evaluate the function at the left end-points and calculate the areas

Fig. 2 shows the graph with 10 rectangles.

Part C. The limit as n ⟶ ∞

Integrate the function.