Evaluate the Riemann sum

Question

f(x) = x² - 4, taking the sample points to be right end points of each subinterval and a = 0, b = 3, and R_6.

Answer 1

f(x) = x^2 - 4

split interval [a,b] up into 6 intervals of length (b-a)/6 = (3-0) / 6 = 1/2

the right sided x values for each subinterval are a + n/2 for n = 1 to 6, so .5, 1, 1.5, 2, 2.5, 3

the width of each interval is delta x = .5

evaluate f at each of the right hand x-values; .5, 1, 1.5, 2, 2.5, 3

obtaining: -3.75, -3.00, -1.75, 0.00, 2.25, 5.00, the heights of the rectangles of width delta x = .5

summing the products of the heights and the widths gives: -0.625

Answer 2

The right end points will be 1/2, 1, 3/2, 2. 5/2, 3

Evaluate f(x) for each of these points.

Multiply the 6 values by 1/2 which is the length of the each interval.

The Riemann sum is the sum of these 6 products.

Note:The "area" from 0 to 2 will be negative and the "area" from 2 to 3 will be positive.

The Riemann sum will NOT be the geometric "area" of the figure defined by the problem!