# Prove that area under curve by rectangle is less than integration

• basil
In summary, the conversation is discussing the task of comparing the area under the curve y=x^2 to rectangles beneath it, specifically in order to prove that the sum of 1/2 + 1/3 + ...+ 1/n is less than or equal to log(n+1). The participants suggest using Riemann sums and subdividing the interval to prove the strict inequality. The correction is made that the curve should be y=1/x and the conversation ends with gratitude for the helpful explanation.
basil
hi all,

I am suppose to compare the area of curve y=x2 with rectangles beneath that curve to show that,

1/2 + 1/3 + ...+ 1/n < log(n+1)

i believe this some sort of harmonic series. Is there a way around this problem?

Regards

basil said:
I am suppose to compare the area of curve y=x2 [...]

Don't you mean y=1/x?

I think it is y=1/x. So, take an arbitrary interval [0,l] and employ riemann sums: divide it in n subintervals, each delta x= l/n wide. The area of all rectangles of basis delta x and height 1/i(delta x), where i runs from 1 to n, is an approximation downwards of the area under the curve. In particular, it will be less than the area from x=delta x to x=l, which is just log n, plus the area of the first rectangle, that is 1. So, sum(from i=1 to n) 1/i <log n +1 and we are done.

basil said:
hi all,

I am suppose to compare the area of curve y=x2 with rectangles beneath that curve to show that,

1/2 + 1/3 + ...+ 1/n < log(n+1)

i believe this some sort of harmonic series. Is there a way around this problem?

Regards

1/2 + 1/3 + ...+ 1/n < or = log(n+1) is by definition of lower sum. To prove strict inequality subdivide more and show that the sum is bigger.

Oh yes, its

$\frac{1}{x}$

Nevertheless, the explanation was beneficial. Thanks guys.

## 1. How does the area under a curve relate to integration?

The area under a curve can be calculated by using integration. Integration is a mathematical technique that allows us to find the total area under a curve by summing up the areas of infinitely many rectangles that approximate the curve. This is known as the definite integral.

## 2. Why is the area under a curve by rectangle less than integration?

The area under a curve can be approximated by using rectangles of equal width and height. However, this method only gives an estimate of the actual area. Integration, on the other hand, takes into account the infinitesimal width of the rectangles, resulting in a more accurate calculation of the area under the curve. Therefore, the area under the curve by rectangle will always be less than the actual area calculated by integration.

## 3. Can you provide an example of how to prove the relationship between the area under a curve and integration?

Sure. Let's take the curve y = x^2 as an example. To find the area under this curve from x = 0 to x = 2, we can use the formula for definite integral: ∫x^2 dx from 0 to 2. This integral evaluates to 8/3. Now, if we divide the area into 4 rectangles of equal width, we get a lower estimate of 4. Therefore, we can see that the area under the curve by rectangle (4) is less than the actual area calculated by integration (8/3).

## 4. Are there any limitations to using the rectangle method to calculate the area under a curve?

Yes, there are limitations to using the rectangle method. The accuracy of the estimated area depends on the number and size of the rectangles used. As the number of rectangles increases, the estimated area becomes closer to the actual area. However, using too many rectangles can be time-consuming and may not always be feasible.

## 5. What are some real-life applications of understanding the relationship between area under a curve and integration?

Understanding the relationship between area under a curve and integration is essential in various fields, such as physics, engineering, economics, and statistics. It is used to calculate physical quantities like displacement, velocity, and acceleration in motion problems. In economics, it is used to calculate the total profit or loss from a production process. In statistics, it is used to calculate probabilities and expected values in probability distributions.

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