# Estimating the value of an integral

• tangibleLime
In summary, the conversation discussed using a known property of integrals to estimate the value of a given integral. The value of the integral was determined to be 5pi/4, with the function 5/(1+x^2) attaining a maximum at x=0 and a minimum at x=1 over the interval [0,1].
tangibleLime

## Homework Statement

It is known that if m[PLAIN]http://www.webassign.net/images/lteq.giff(x)[ATTACH=full]196966[/ATTACH]M for a[PLAIN]http://www.webassign.net/images/lteq.gifx[ATTACH=full]196967[/ATTACH]b, then the following property of integrals is true.

Use this property to estimate the value of the given integral.

____________ <=
<= _____________

## The Attempt at a Solution

Honestly, I don't really know what it wants for a solution. I put in two numbers that satisfied the inequality, but it marked it wrong, so I assume it wants some sort of expression. The value of the integral itself is 5pi/4, but other than that I am at a complete loss for how to even begin.

Nevermind, I figured it out... I was overcomplicating it. I didn't realize they were using M and m as a function... I thought they were multiplying it. So m(1) <= 5/(1+1^2) = 5/2, and so-forth.

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They were multiplying it. Over the interval [0,1], the function 5/(1+x2) attains a maximum at x=0 and a minimum at x=1, so you can use the values m=f(1) and M=f(0) in the inequality.

## 1. What is an integral and why is it important?

An integral is a mathematical concept that represents the area under a curve on a graph. It is important because it allows us to calculate the total quantity or value of a continuous function, which is useful in many real-world applications such as calculating volume, distance, and probability.

## 2. How do you estimate the value of an integral?

The most common method for estimating the value of an integral is by using numerical integration techniques such as the trapezoidal rule or Simpson's rule. These methods involve dividing the area under the curve into smaller sections and approximating the area of each section using simple geometric shapes.

## 3. What are the limitations of estimating integrals?

Estimating integrals can be limited by the accuracy of the numerical integration method used. Some methods may not be able to accurately estimate the value of integrals for complex functions with irregular curves. Additionally, the estimated value may also be affected by the number of subdivisions used in the calculation.

## 4. How does the choice of interval affect the estimation of an integral?

The choice of interval, or the range of values over which the integral is being calculated, can greatly affect the accuracy of the estimation. If the interval is too large, the estimated value may not be accurate enough. Conversely, if the interval is too small, the calculation may take longer to complete and may not significantly increase the accuracy of the estimation.

## 5. Are there any other methods for estimating integrals?

Yes, there are other methods for estimating integrals, such as Monte Carlo integration, which involves using random numbers to estimate the value of an integral. This method can be useful for complex functions with irregular curves, but it may also require a large number of iterations to achieve an accurate estimation.

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