Sketch g(x) = x-4. Then using summation notation, write an expression for A(g, 5 ≤ x ≤ 7) using five right endpoint rectangles

Answer 1

graphing and using geometry

the area = bh/2 = 3(3/2 = 4.5

the area of a right triangle, base =7-4=3 and height=3

G(x) = integral of g(x) =

integral of x-4 = x^2/2 -4x (ignore the constant term)

evaluated from 4 to 7

= 49/2-28 -(8-16) = 24.5 -28 +8 = 4.5

"integral" sign is a version of a summation sign, an "S" shape

5 right end points mean base of (7-4)/5 = 3/5 for each rectangle

heights are 3/5 6/5, 9/5 12/5 3

Sum of areas = (3/5)(3+6+9+12+15)/5 = 5.4 right ends overestimate the area

left ends underestimate the area = (3/5)(0+3+6+9+12)/5 = 3.6

their average = actual area

(3.6+5.4)/2 = 4.5