Calculus Integrals

Question

Given the table below for selected values of f(x), use 6 left rectangles to estimate the value of

.

x | 1, 3, 4, 6, 7, 9, 10

f(x) | 4, 8, 6, 10, 10, 12, 16

Which of the following values would be obtained using 5 midpoint rectangles of equal width to estimate ?

Answer 1

Using left endpoints, ∫_{(from 1 to 10)}f(x)dx ≈ f(1)(3-1) + f(3)(4-3) + f(4)(6-4) + f(6)(7-6) + f(7)(9-7) + f(9)(10-9)

= 8 + 8 + 12 + 10 + 20 + 12 = 70

Using the Midpoint Rule with 5 rectangles of equal width,

∫_{(from 0 to 1)}x²dx ≈ (0.1)²(0.2) + (0.3)²(0.2) + (0.5)²(0.2) + (0.7)²(0.2) + (0.9)²(0.2) = 0.33