Evaluate the Riemann sum

Question

If f(x) = 4x² − 5x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.

Answer 1

Divide the space into equal sections of 6 (n = 6) so each section is 0.5 wide. Calculate the function values on each section. Use the right sides to create a series of 6 rectangular boxes and use geometry to calculate the area.

A = (0.5)(-1.5) + (0.5)(-1) + (0.5)(1.5) + (0.5)(6) + (0.5)(12.5) + (0.5)(21)

or A = 0.5(-1.5 + -1 + 1.5 + 6 + 12.5 + 21)

A = 19.25

so ₀∫³4x² - 5x dx ≈ 19.25