# Integration-Lower/Upper sum, Partition problem

• SolidSnake
In summary: Your Name]In summary, we need to show that for any partition P, the integral of a function f on [a,b] minus the lower sum of f with the partition P is less than or equal to [f(b) - f(a)]Δx. To do this, we use the fact that f is increasing on [a,b] and manipulate the inequality to obtain the desired result.
SolidSnake

## Homework Statement

Suppose f is increasing and continuous on [a,b]. Show for any partition P,

$$\int^b_a f(x)dx - L_f(P) \leq [f(b) - f(a)] \Delta x$$

## Homework Equations

Not sure if there are any . but for people unfamiliar with this notation:
$$L_f(P)$$ = Lower sum for f with the partition P
$$U_f(P)$$ = Upper sum for f with the partition P

## The Attempt at a Solution

If you defined $$P = \{x_0, x_1, x_2,...,x_n_-_1, x_n\}$$ $$a = x_0$$ and $$b = x_n$$

I also know that
$$L_f(P) \leq \int^b_a f(x)dx \leq U_f(P)$$
If i subtract The lower sum from both sides i get,

$$0 \leq \int^b_a f(x)dx - L_f(P) \leq U_f(P) - L_f(P)$$

which gives me the left side of what I'm trying to show. Now the problem arises trying to make the right side look like what it should. When the partitions are of equal size, the lower and upper sum's become a telescoping sum (I know this from a previous question), but in the given case, where the partitions can be of any size i can't find a way to make it look like the right side of the inequality. I've expanded both the lower and upper sums but was unable to cancel anything or find a way to manipulate it to make it look like $$[f(b) - f(a)] \Delta x$$.

Any help would be great.

Last edited:

Firstly, we need to recall the definition of a Riemann sum for a function f on the interval [a,b] with a partition P = {x0, x1, x2, ..., xn-1, xn}. The Riemann sum is given by:

R(f,P) = ∑f(xi)(xi+1 - xi)

Now, we know that for any partition P, the lower sum Lf(P) is less than or equal to the integral of f on [a,b] which is also less than or equal to the upper sum Uf(P). Therefore, we can write:

Lf(P) ≤ ∫f(x)dx ≤ Uf(P)

Subtracting Lf(P) from both sides, we get:

0 ≤ ∫f(x)dx - Lf(P) ≤ Uf(P) - Lf(P)

Now, we need to make the right side look like [f(b) - f(a)]Δx. For this, we will use the fact that f is increasing on [a,b]. This means that for any x in [a,b], f(x) ≥ f(a) and f(x) ≤ f(b). Therefore, we can write:

f(a) ≤ f(x) ≤ f(b)

Multiplying both sides by Δx, we get:

f(a)Δx ≤ f(x)Δx ≤ f(b)Δx

Now, we can substitute this in the inequality we obtained above:

0 ≤ ∫f(x)dx - Lf(P) ≤ Uf(P) - Lf(P)
≤ f(b)Δx - f(a)Δx = [f(b) - f(a)]Δx

Hence, we have shown that for any partition P, we have:

0 ≤ ∫f(x)dx - Lf(P) ≤ [f(b) - f(a)]Δx

which is the required result.

I hope this helps you understand the problem better. If you have any further questions, please do not hesitate to ask.

## 1. What is Integration?

Integration is a mathematical concept that involves finding the area under a curve. It is used to solve problems related to finding the total value of a quantity that keeps changing over a given interval.

## 2. What are Lower and Upper Sums?

Lower and Upper Sums are two methods used to approximate the area under a curve during the process of integration. Lower sums use rectangles with their heights determined by the lowest point on each interval, while upper sums use rectangles with their heights determined by the highest point on each interval.

## 3. How is Partitioning used in Integration?

Partitioning is the process of dividing the interval on which the area is to be calculated into smaller subintervals. These subintervals are used in both lower and upper sums to approximate the area under the curve.

## 4. What is the Partition Problem?

The Partition Problem is the task of finding the best partition of an interval to use in the integration process. This involves determining the number of subintervals and their size in order to get the most accurate approximation of the area under the curve.

## 5. How is Integration-Lower/Upper sum used in real-life applications?

Integration-Lower/Upper sum is used in various fields such as physics, engineering, and economics to solve problems related to finding the total value of a quantity that changes over time. It is also used in calculating areas, volumes, and other quantities in real-life scenarios.

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