Integral
In calculus, an integral is the space under a graph of an equation (sometimes said as "the area under a curve"). An integral is the reverse of a derivative, and integral calculus is the opposite of differential calculus. A derivative is the steepness (or "slope"), as the rate of change, of a curve. The word "integral" can also be used as an adjective meaning "related to integers".
The symbol for integration, in calculus, is: [math]\displaystyle{ \textstyle \int_{\,}^{\,} }[/math] as a tall letter "S".[1][2][3]
Integrals and derivatives are part of a branch of mathematics called calculus. The link between these two is very important, and is called the fundamental theorem of calculus.[4] The theorem says that an integral can be reversed by a derivative, similar to how an addition can be reversed by a subtraction.
Integration helps when trying to multiply units into a problem. For example, if a problem with rate, [math]\displaystyle{ \left(\tfrac{\text{distance}}{\text{time}}\right) }[/math], needs an answer with just distance, one solution is to integrate with respect to time. This means multiplying in time to cancel the time in [math]\displaystyle{ \left(\tfrac{\text{distance}}{\text{time}}\right)\times\text{time} }[/math]. This is done by adding small slices of the rate graph together. The slices are close to zero in width, but adding them together indefinitely makes them add up to a whole. This is called a Riemann sum.
Adding these slices together gives the equation that the first equation is the derivative of. Integrals are like a way to add many tiny things together by hand. It is like summation, which is adding [math]\displaystyle{ 1+2+3+4....+n }[/math]. The difference with integration is that we also have to add all the decimals and fractions in between.[4]
Another time integration is helpful is when finding the volume of a solid. It can add two-dimensional (without width) slices of the solid together indefinitely—until there is a width. This means the object now has three dimensions: the original two and a width. This gives the volume of the three-dimensional object described.
Methods of Integration
Antiderivative
By the fundamental theorem of calculus, the integral is the antiderivative.
If we take the function [math]\displaystyle{ 2x }[/math], for example, and anti-differentiate it, we can say that an integral of [math]\displaystyle{ 2x }[/math] is [math]\displaystyle{ x^2 }[/math]. We say an integral, not the integral, because the antiderivative of a function is not unique. For example, [math]\displaystyle{ x^2+17 }[/math] also differentiates to [math]\displaystyle{ 2x }[/math]. Because of this, when taking the antiderivative a constant C must be added. This is called an indefinite integral. This is because when finding the derivative of a function, constants equal 0, as in the function
- [math]\displaystyle{ f(x) = 5x^2 + 9x + 15\, }[/math].
- [math]\displaystyle{ f'(x) = 10x + 9 + 0\, }[/math]. Note the 0: we cannot find it if we only have the derivative, so the integral is
- [math]\displaystyle{ \int (10x + 9)\, dx = 5x^2 + 9x + C }[/math].
Simple Equations
A simple equation, such as [math]\displaystyle{ y = x^2 }[/math], can be integrated with respect to x using the following technique. To integrate, you add 1 to the power x is raised to, and then divide x by the value of this new power. Therefore, integration of a normal equation follows this rule: [3][math]\displaystyle{ \int_{\,}^{\,} x^n dx = \frac{x^{n+1}}{n+1} + C }[/math]
The [math]\displaystyle{ dx }[/math] at the end is what shows that we are integrating with respect to x, that is, as x changes. This can be seen to be the inverse of differentiation. However, there is a constant, C, added when integrating. This is called the constant of integration.[1] This is required because differentiating a number results in zero, therefore integrating zero (which can be put onto the end of any integrand) produces a constant, C. The value of this constant would be found by using given conditions.
Equations with more than one terms are simply integrated by integrating each individual term:
[math]\displaystyle{ \int_{\,}^{\,} x^2 + 3x - 2 dx = \int_{\,}^{\,} x^2 dx + \int_{\,}^{\,} 3x dx - \int_{\,}^{\,} 2 dx = \frac{x^3}{3} + \frac{3x^2}{2} - 2x + C }[/math]
Integration involving e and ln
There are certain rules for integrating using e and the natural logarithm. Most importantly, [math]\displaystyle{ e^x }[/math] is the integral of itself (with the addition of a constant of integration): [3][math]\displaystyle{ \int_{\,}^{\,}e^{x} dx = e^{x} + C }[/math]
The natural logarithm, ln, is useful when integrating equations with [math]\displaystyle{ 1/x }[/math]. These cannot be integrated using the formula above (add one to the power, divide by the power), because adding one to the power produces 0, and a division by 0 is not possible. Instead, the integral of [math]\displaystyle{ 1/x }[/math] is [math]\displaystyle{ \ln x }[/math]: [math]\displaystyle{ \textstyle \int_{\,}^{\,}\frac{1}{x} dx = \ln x + C }[/math][3]
In a more general form: [math]\displaystyle{ \int_{\,}^{\,}\frac{f'(x)}{f(x)} dx = \ln {|f(x)|} + C }[/math]
The two vertical bars indicated a absolute value; the sign (positive or negative) of [math]\displaystyle{ f(x) }[/math] is ignored. This is because there is no value for the natural logarithm of negative numbers.
Properties
Sum of functions
The integral of a sum of functions is the sum of each function's integral. that is,
- [math]\displaystyle{ \int\limits_{a}^{b} [f(x) + g(x)]\, dx = \int\limits_{a}^{b} f(x)\, dx + \int\limits_{a}^{b} g(x)\, dx }[/math].
The proof of this is straightforward: The definition of an integral is a limit of sums. Thus
- [math]\displaystyle{ \int\limits_{a}^{b} [f(x) + g(x)]\, dx = \lim_{n \to \infty} \sum_{i=1}^n \left(f(x_i^*) + g(x_i^*)\right) }[/math]
- [math]\displaystyle{ = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) + \sum_{i=1}^n g(x_i^*) }[/math]
- [math]\displaystyle{ = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) + \lim_{n \to \infty} \sum_{i=1}^n g(x_i^*) }[/math]
- [math]\displaystyle{ = \int\limits_{a}^{b} f(x)\, dx + \int\limits_{a}^{b} g(x)\, dx }[/math]
Note that both integrals have the same limits.
Constants in integration
When a constant is in an integral with a function, the constant can be taken out. Further, when a constant c is not accompanied by a function, its value is c * x. That is,
- [math]\displaystyle{ \int\limits_{a}^{b} cf(x)\, dx = c \int\limits_{a}^{b} f(x)\, dx }[/math] and
This can only be done with a constant.
- [math]\displaystyle{ \int\limits_{a}^{b} c\, dx = c(b-a) }[/math]
Proof is again by the definition of an integral.
Other
If a, b and c are in order (i.e. after each other on the x-axis), the integral of f(x) from point a to point b plus the integral of f(x) from point b to c equals the integral from point a to c. That is,[3]
- [math]\displaystyle{ \int\limits_{a}^{b} f(x)\, dx + \int\limits_{b}^{c} f(x)\, dx = \int\limits_{a}^{c} f(x)\, dx }[/math]
- if they are in order. (This also holds when a, b, c are not in order if we define
- [math]\displaystyle{ \textstyle \int\limits_{a}^{b} f(x) \,dx= -\int\limits_{b}^{a} f(x)\, dx }[/math].)
- [math]\displaystyle{ \int\limits_{a}^{a} f(x)\, dx = 0 }[/math]
- This follows the fundamental theorem of calculus (FTC): [math]\displaystyle{ F(a)-F(a) = 0 }[/math].
- [math]\displaystyle{ \int\limits_{a}^{b} f(x)\, dx = -\int\limits_{b}^{a} f(x)\, dx }[/math]
- Again, following the FTC: [math]\displaystyle{ F(b)-F(a) = -[F(a)-F(b)] }[/math].
Integral Media
A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area
Approximations to integral of √x from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions
Demonstrates a step towards the Lebesgue integration of a function using a simple function.
The improper integral\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pihas unbounded intervals for both domain and range.
Double integral computes volume under a surface z=f(x,y)
A line integral sums together elements along a curve.
The definition of surface integral relies on splitting the surface into small surface elements.
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method (at varying piece count), Gaussian quadrature
Related pages
References
- ↑ 1.0 1.1 List of Calculus and Analysis Symbols (in en-US). Math Vault (2020-05-11). Retrieved 2020-09-18.
- ↑ Weisstein, Eric W.. Integral (in en). mathworld.wolfram.com. Retrieved 2020-09-18.
- ↑ 3.0 3.1 3.2 3.3 3.4 Integral calculus - Encyclopedia of Mathematics. encyclopediaofmath.org. Retrieved 2020-09-18.
- ↑ 4.0 4.1 Barton, David. Delta Mathematics (2003)Pearson Education. ISBN 0-582-54539-0.