If I have a table that shows x inputs: 0,3,6,9,12, and 15 and then f(x) for them: 63,62,59,52,38, and 10, how can I estimate the integral if I know it is a left Riemann sum covering the area of 0 to 15? What about if right Riemann sum?

since the interval width is 3, the left sum is

3(f(0)+f(3)+f(6)+f(9)+f(12)) = 3(63+62+59+52+38) = 822
the right sum is
3(f(3)+f(6)+f(9)+f(12)+f(15)) = 3(62+59+52+38+10) = 663

Did you know?

Did you know that the left Riemann sum is a method used to estimate the area under a curve, based on dividing the region into small rectangles and calculating their combined areas? In the given example, if we have a table showing inputs x and their corresponding f(x) values, and we know it is a left Riemann sum covering the interval from 0 to 15, we can estimate the integral by multiplying the width of each rectangle (which is equal to the distance between consecutive x values) by the corresponding f(x) value and summing them up. On the other hand, did you know that the right Riemann sum is another method to estimate the area under a curve, but with a slightly different approach? If we use a right Riemann sum to estimate the integral in the same scenario, we would use the f(x) value at the right endpoint of each rectangle. This means that we would estimate the area by multiplying the width of each rectangle by the f(x) value corresponding to the x value at its right endpoint and summing them up. Both left and right Riemann sums are useful techniques for approximating integrals, allowing us to gain insights into the overall behavior of functions and the areas they encompass.