Incalculus, anintegral is thespace under agraph of anequation (sometimes said as "thearea under a curve"). An integral is the reverse of aderivative, and integral calculus is the opposite ofdifferential calculus. Aderivative is the steepness (or "slope"), as therate of change, of a curve. The word "integral" can also be used as anadjective meaning "related tointegers".
Thesymbol for integration, in calculus, is: as a tall letter "S".[1][2][3]
Integrals and derivatives are part of a branch ofmathematics calledcalculus. The link between these two is very important, and is called thefundamental 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 asubtraction.
Integration helps when trying tomultiply units into a problem. For example, if a problem withrate,, 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. 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 aRiemann sum.
Adding these slices together gives theequation that the first equation is the derivative of. Integrals are like a way to add many tiny things together by hand. It is likesummation, which is adding. The difference with integration is that we also have to add all thedecimals andfractions in between.[4]
Another time integration is helpful is when finding thevolume of asolid. It can addtwo-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 thethree-dimensional object described.
If we take the function, for example, and anti-differentiate it, we can say that an integral of is. We sayan integral, notthe integral, because the antiderivative of a function is not unique. For example, also differentiates to. Because of this, when taking the antiderivative a constant C must be added. This is called an indefinite integral. This is because when finding thederivative of a function, constants equal 0, as in the function
.
. Note the 0: we cannot find it if we only have the derivative, so the integral is
A simple equation, such as, can be integrated with respect to x using the following technique. To integrate, you add 1 to the powerx is raised to, and then dividex by the value of this new power. Therefore, integration of a normal equation follows this rule:[3]
The at the end is what shows that we are integratingwith respect to x, that is, asx changes. This can be seen to be theinverse ofdifferentiation. However, there is a constant, C, added when integrating. This is called the constant of integration.[1] This is required because differentiating an integer results inzero, therefore integrating zero (which can be put onto the end of any integrand) produces an integer, C. The value of this integer would be found by using given conditions.
Equations with more than one terms are simply integrated by integrating each individual term:
There are certain rules for integrating usinge and thenatural logarithm. Most importantly, is the integral of itself (with the addition of a constant of integration):[3]
The natural logarithm, ln, is useful when integrating equations with. 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 is:[3]
In a more general form:
The two vertical bars indicated aabsolute value; the sign (positive or negative) of is ignored. This is because there is no value for the natural logarithm of negative numbers.
When a constant is in an integral with a function, the constant can be taken out. Further, when a constantc is not accompanied by a function, its value isc *x. That is,
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]
if they are in order. (This also holds when a, b, c are not in order if we define
