Question

Use the rectangles to approximate the area of the region. f(x) = -x + 11 [1, 11] y 10 8 6 2 2 4 6 8 10 10 Х Give the exact area obtained using a definite integral. 10 x Need Help? Read it Watch It Talk to a Tutor

Answer 1

-x+1) f(x)= Height of Rectangle = f (2.25 + (K-1) 2.5) [Base of earth Rectangle "} stolsk = 4 2.5 integer from graph is given Thus approximation rectangle by by A = f(2-25)x2.5 + f(2:25 +2.5) x2.5 + f(2.25 +5 )x2-5 + f(2-25 4- 7.5) Las = 2.5 [f (2.25) + f(4.75) + f(7.25)+f69.15) - = 2.5 [-2.25-475 - 7.25 -5.75+44] 2.5 [-24 +44 S.S. 일 2.5.0 20 A 50 Actual value = Stetida I = [<< +112], Exact Area > I -50

from graph f(x)= 25-2², [55] Buse of each rectangle 5+5 = 2.5 lo 4 - Height of where ISK < 4 kt rectangle f (-3.75+(K-1) 2:5) and K is integer. This approximation by rectangle is given by A FC-3-75) *2.5 + f 4-1.25)¥2.5 + f(1.25)X2.57 it f (3:25) *2.5 - 2.5 [100-( 3.75+ 5.25%+1.25°+3.753] 205 [ 100-31.25] - 2.5 x 68.75 5 14 = 171.875 Actual Area is given by (25-x²) dx = [125-125 + 125 - 13] = 5. 252 - x 3 -5 I = 55 5oo 1 3 Exact Area I=500 3

Bra th integer f(x) = 49-2x² El,o 0-(1) - 13 base of each of 5 rectangles Height of Ith rectangle af rectangle = f(- 9 / 0 + (k-1) + ), IEK85 This approximated area is given by A = f(-0.9)*/3 + 460.7 x 12 + f(-0.5x18+ + f(-0.3)* 1/3 + f(-0.1)* 15 + [5x49 - (0.9+0.7 +0.5+0.3 70 0.13) 5 . [1964 1965) 243.35 5 os * *4.35 A = 48.67 18.87