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sketch the graph of the function y=20x-x^ ^ ^2 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 :

Y = 20x - x^2. (Parabola curve)

Area of curve in [ 0,20 ]

Part A. 5 rectangles

Each rectangle measures

Base1 =20/5 = 4

Height1 = 20•4- 4^2 = 64

Base2= 4

Height2 = 20•8 - 8^2 = 96

Base3= 4

Height3= 20•12 -12^2 = 96

Base4 = 4

Height4= 20•16 - 16^2 = 64

Base5= 4

Height5 = 20•20 - 20^2 = 0

Then area under curve is = 4x (64+96+96+64+0)

. = 4 x 320 = 1280

Part B) 10 rectangles

Then

Base of rectangles= 20/10 = 2

Heights of rectangles= 20•2 - 2^2 = 36

. = 20•4 - 4^2 = 64

. = 20•6 - 6^2 = 84

. = 20•8 - 64 = 96

. = 20•10 - 100= 100

. = 20•12 - 144= 96

. = 20•14 - 196 = 84

. = 20•16 - 256 = 64

. = 20•18 - 324 = 36

Then now AREA IS = 2x (36+64+84+96+100+96+84+64+36)

. = 2x 660

. = 1320

Part C) Area under the curve, with infinite rectangles

Base of rectangles = 20/X

X goes to infinity

Height of rectangles = 20•(20/x) + (20/x)^2

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