Can the upper and lower sums be equal?

Since the supremum over some set can’t be smaller than the infimum over the same set, equality of the upper and lower sums implies that the supremum and infimum are equal, and thus the function itself must map the entire domain to a constant.

Which conditions are necessary to explain that the upper and lower integrals of a bounded function exist?

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , , , and be bounded.

What are the three types of Riemann sums?

There are three basic types of Riemann sum that could show up on the Calculus BC exam.

• Right endpoint sum.
• Left endpoint sum.
• Midpoint Rule.

What is upper and lower Riemann sum?

To form the Upper Riemann Sum, we use the maximum height of the rectangle on each subinterval. The Lower Riemann Sum uses the minimum height of the rectangle on each subinterval. Thus, L(P, f) = ∆x(f(−. 6) + f(−.

What is the necessary and sufficient condition for Riemann integrability of a bounded function f?

A function is called Riemann integrable if and only if it is bounded and continuous almost everywhere on its domain. However, I have read that the following two statements are also true: a) If f is continuous then f is Riemann integrable. b) If f is bounded then f is Riemann integrable.

What is lower and upper integral?

The lower Darboux integral of f is. In some literature an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively. and like Darboux sums they are sometimes simply called the lower and upper integrals. If Uf = Lf, then we call the common value the Darboux integral.

What is upper Riemann integral?

The integral as the area of a region under a curve. A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function.

What is upper and lower sum?

The total area of the inscribed rectangles is the lower sum, and the total area of the circumscribed rectangles is the upper sum. By taking more rectangles, you get a better approximation.

What is a lower Riemann sum?

the lower sum is the sum of the areas of rectangles which are always lower than. the graph of the function; see Figure 1.

What is a specific Riemann sum?

When the points x i ∗ are chosen randomly, the sum ∑ i = 1 n f (x i ∗) Δ x i is called a Riemann Sum and will give an approximation for the area of R that is in between the lower and upper sums. The upper and lower sums may be considered specific Riemann sums.

How can I approximate the area under a Riemann curve?

The following Exploration allows you to approximate the area under various curves under the interval [ 0, 5]. You can create a partition of the interval and view an upper sum, a lower sum, or another Riemann sum using that partition. The Exploration will give you the exact area and calculate the area of your approximation.

What is definition of the Riemann integral?

Definition of the Riemann integral. We say that two intervals are almost disjoint if they are disjoint or intersect only at a common endpoint. For example, the intervals [0,1] and [1,3] are almost disjoint, whereas the intervals [0,2] and [1,3] are not. Definition 1.1.

How do I create a partition of the interval?

You can create a partition of the interval and view an upper sum, a lower sum, or another Riemann sum using that partition. The Exploration will give you the exact area and calculate the area of your approximation. To create a partition, choose which type of sum you would like to see and click the mouse between the partition labels x 0 and x 1 .