Question

HELLP 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. part a Use five rectangles to approximate the area under the curve. part b Use 10 rectangles to approximate the area under the curve. part c 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.

Answer 1

Very Fascinating man

Answer 2

Answer:

Part a. 1400 unit²

Part b  1240 unit²

Part c 1333.33 unit²

Step-by-step explanation:

The given function is 20·x - x²

Part a.

The dimensions of the five rectangles are;

20 × 20 + 20 × 18 + 20 × 16 + 20 × 10 + 20 × 6 = 1400 unit²

Part b.

The dimensions of the ten rectangles are;

10×20 + 10×18 + 10×16 + 10×16 + 10×14 + 10×12 + 10 × 10 + 10 × 8 + 10 × 6 + 10 ×4 = 1240 unit²

Part c.

To find the area with an infinite number of rectangles, we can use limits similar to integration;

$\int _{0}^{20}\left (20\cdot x - x^{2} \right )dx$

Which gives;

$\int _{0}^{20}\left (20\cdot x - x^{2} \right )dx = \left [\dfrac{20}{2}\cdot x^{2} - \dfrac{x^{3}}{3} \right ]_{0}^{20} = \left [10\cdot x^{2} - \dfrac{x^{3}}{3} \right ]_{0}^{20}$

10×20^2 - (20^3)/3 = 4000/3 = 1333.3333