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

Numerical Answers Expected!

Answer 1

x f(x)

1 4

3 8

4 6

6 10

7 10

9 12

10 16

Your ragged-edged estimate is 2*4 + 1*8 + 2*6 + 1*10 +2*10 + 1*12 = 8+8+12+10+20+12 = 70

Using rectangles of equal width, (10-1)/6 = 3/2, so the partition is: 1, 2.5, 4, 5.5, 7, 8.5

We need f(2.5), f(5.5), f(8.5) via lineari-zation.

on [1,3], the slope is 4/2 = 2.

the intecept B = y - mx = 4 - (2)(1) = 2<--- using the LEFT point

y = 2x +2 ---> so when x = 2.5, the function is ABOUT 2(2.5) +2 = 5+2 = 7.

on [4,6], the slope is also m=2.

intercept B = y - mx = 6 - 4(2) = 6 - 8 = -2

y = 2x - 2 --> so when x = 5.5, the function is ABOUT 2(5.5) - 2 = 11 - 2= 9

on [7,9], the slope is m=1.

intercept B = 10 - (1)(7) = 3

y = x+3 --> so when x=8.5, the function is ABOUT 11.5

so for step size h=1.5, factored out, the area is approximately

1.5 * { f(1) + f(2.5) + f(4) + f(5.5) + f(7) + f(8.5) }

1.5 * [ 4 + 7 + 6 + 9 + 10 + 11.5] =

1.5 * 47.5 = 71.25