Mastering Riemann Sums: An Essential Guide To Approximating Areas Under Functions

Riemann Sums

(b-a)/n * (sum of y values)n = number of rectangles you make,

Riemann sums are a way to approximate the area between a function and the x-axis on a given interval. This is done by dividing the interval into equally sized subintervals and evaluating the function at different points within each subinterval.

There are three main types of Riemann sums: left, right, and midpoint. In a left Riemann sum, the function is evaluated at the left endpoint of each subinterval. In a right Riemann sum, the function is evaluated at the right endpoint of each subinterval. In a midpoint Riemann sum, the function is evaluated at the midpoint of each subinterval.

To find the Riemann sum of a function on a given interval, we must first determine the width of each subinterval, which is given by:

Δx = (b – a) / n

where Δx is the width of each subinterval, b is the upper bound of the interval, a is the lower bound of the interval, and n is the number of subintervals.

Next, we must determine the points at which to evaluate the function within each subinterval. For example, in a left Riemann sum, we would evaluate the function at the left endpoint of each subinterval, which would be:

a, a + Δx, a + 2Δx, …, a + (n – 1)Δx

Once we have determined the points at which to evaluate the function, we can then calculate its value at each of these points and multiply each by the width of the corresponding subinterval. The sum of these products gives us the Riemann sum of the function on the given interval.

As the number of subintervals increases, the Riemann sum becomes a more accurate approximation of the area between the function and the x-axis. In the limit as n approaches infinity, the Riemann sum becomes the exact value of this area and is called the definite integral of the function over the given interval.

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