Riemann Sums

A Riemann Sum is a method for approximating the total area under a curve by dividing it into smaller rectangles, calculating the area of each rectangle, and summing these areas.

$$\Delta x$$ represents the width of each sub-interval in the partition of the interval over which the integral is being approximated.

$$\Delta x = \frac{b-a}{n} = \frac{4-1}{n} = \frac{3}{n}$$ where a=1 and b=4.

The Riemann Sum using right endpoints is given by $$R_n = \sum_{k=1}^{n} f(x_k) \Delta x$$ where $$x_k = a + k \Delta x$$ .

Left Riemann Sums use the left endpoints of the sub-intervals to calculate the height of the rectangles, while right Riemann Sums use the right endpoints.

A partition refers to the division of the interval into smaller sub-intervals for the purpose of approximating the area under the curve.

The limit of Riemann Sums as n approaches infinity gives the exact value of the definite integral, representing the total area under the curve.

If the function is increasing, the left Riemann Sum will underestimate the actual value of the integral.

The height of rectangles in a left Riemann Sum is given by $$f(x_k)$$ where $$x_k = a + (k-1) \Delta x$$ .

Riemann Sums provide a method to approximate definite integrals, and as the number of sub-intervals increases, the Riemann Sum approaches the value of the definite integral.