Approximate the area under the curve y=x^2+5 from x=2 to x=5 using a left sum with 6 subdivisions.

Question

Approximate the area under the curve ² from  to  using a left sum with 6 subdivisions.

The area is approximately _____

Answer 1

integral of x^2 +5 = x^3/3 +5x

evaluated from 2 to 5

= 125/3 + 25 -[8/3 +10]

= 125/3 +25 - 8/3 -10

= 117/3 +15

= 39 + 15 = 54 = exact area

Reiman left sums should be < 54

6 divisions mean the base of each rectangle = (5-2)/6 = 1/2

heights of each of the 6 rectangles are: f(2), f(2.5), f(3), f(3.5), f(4), and f(4.5)

= 9, 11.25, 14, 17.25, 21 and 25.25

sum them = 20.25 + 31.25 + 46.25 = 97.75

multiply be the base = (1/2)(97.75) = 48.875 < 54