Suppose f(x) = (x^2)/9. The rectangles in the graph on the left illustrate a left endpoint Riemann sum for f(x) on the interval 2 ≤ x ≤ 4.

Question

Suppose f(x) = (x^2)/9.

Left endpoint Riemann sum		Right endpoint Riemann sum

(a) The rectangles in the graph on the left illustrate a left endpoint Riemann sum for f(x) on the interval 2 ≤ x ≤ 4. The value of this left endpoint Riemann sum is (?), and it is an underestimate of the area of the region enclosed by y = f(x), the x-axis, and the vertical lines x = 2 and x = 4.

(b) The rectangles in the graph on the right illustrate a left endpoint Riemann sum for f(x) on the interval 2 ≤ x ≤ 4. The value of this right endpoint Riemann sum is (?), and it is an overestimate of the area of the region enclosed by y = f(x), the x-axis, and the vertical lines x = 2 and x = 4.

Answer 1

(a) L₄ = 1/9·1/2(2² + 2.5² + 3² + 3.5²) = 1.75

(b) R₄ = 1/9·1/2(2.5² + 3² + 3.5² + 4²) = 29/12 = 2.42