(a) Find an approximation to the integral from 0 to 4 (x^2 − 3x) dx using a Riemann sum with right endpoints and n = 8.

Question

(a) Find an approximation to the integral from 0 to 4 (x^2 − 3x) dx using a Riemann sum with right endpoints and n = 8.

Answer 1

Your question does not say what n is but I will assume it means the number of rectangles in your Riemann sum.

We divide [0,4] into 8 subintervals with right endpoints;

The widths of those smaller intervals would be delta_x or .5

We get the following right subinterval endpoints: 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0

Remember, this is a right Riemann sum.

Now we evaluate the function x^2-3x at those right endpoints and we get the heights of our 8 little rectangles of -1.25, -2.00, -2.25, -2.00, -1.25, 0.00, 1.75, 4.00; next we multiply the width of each rectangle times its height and we get the areas of our 8 rectangles: -0.625 ,-1.000, -1.125, -1.000, -0.625, 0.000, 0.875, 2.000;

and finally, we add those little guys all up and get our desired Reimann sum of -1.5.

This of course is an approximation to the integral of the same function with the same limits which is the limit of Riemann sum if we let delta_x go to 0 and the number of rectangles therefore approach infinity.