Integral from negative infinity to infinity

MHB
Thread starter juantheron
Start date Jul 26, 2012
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Infinity Integral Negative

In summary, the integral from negative infinity to infinity is a mathematical concept used to find the area under a curve that extends infinitely in both directions. It is calculated using the fundamental theorem of calculus and only exists when the function being integrated converges to a finite value. This concept is significant in various real-world applications, such as finding total area or volume and solving differential equations. The integral can also have a negative value if the function being integrated has negative values over certain intervals.

Jul 26, 2012

juantheron

$\displaystyle (1)\;\; \int_{-\infty}^{\infty}\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}dx$

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Jul 27, 2012

CaptainBlack

\[\int_{-\infty}^{\infty}\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}dx=\pi\]

Now can you tell us in what context this arises.

CB

1. What is the meaning of "integral from negative infinity to infinity"?

The integral from negative infinity to infinity is a mathematical concept used to find the area under a curve that extends infinitely in both directions. It represents the sum of infinitely many infinitesimal values between negative infinity and positive infinity.

2. How is the integral from negative infinity to infinity calculated?

The integral from negative infinity to infinity can be calculated using the fundamental theorem of calculus. This involves finding the antiderivative of the function and evaluating it at the limits of negative infinity and positive infinity.

3. Is the integral from negative infinity to infinity always defined?

No, the integral from negative infinity to infinity is not always defined. It only exists when the function being integrated converges to a finite value as the limits approach negative infinity and positive infinity.

4. What is the significance of the integral from negative infinity to infinity in real-world applications?

The integral from negative infinity to infinity is used in various fields of science and engineering to find the total area or volume of a continuous system. It is also used to calculate probabilities in statistics and to solve differential equations in physics and engineering.

5. Can the integral from negative infinity to infinity have a negative value?

Yes, the integral from negative infinity to infinity can have a negative value if the function being integrated has negative values over certain intervals. In this case, the negative value represents the area below the x-axis on the interval where the function is negative.