Integration
The integration is the inverse process of differentiation and is also commonly known as the antiderivative. Calculus defines integration as the algebraic method of determining the integral of a function at any point on a graph of a function.
The integration and differentiation combinedly form the basic theorem of calculus. Integration in calculus is mainly implied for calculation for the area under the curve.
Due to the fact that the area under the curve is not a perfect shape for which the area can be computed, therefore the concept that fully satisfies and determines what area under the curve is integral.
Other than being employed to determine the surface areas of curved bodies the integration is also used to calculate the volume of objects. Moreover, there are several real-world applications of the process of integration.
Double Integration
When applied to two dimensions, the double integral is an extension of the definite integral. However, the concept of a definite integral was introduced for calculating areas in the plane.
Thus, the volume of the solid formed as a result of the integrand and contained inside the integration region is represented by the double integral. When dealing with double integrals, the notation employed is
In order to properly define double integrals in mathematical terms, f(x,y) is a two-variable function whose integral over a region R is called a double integral. It is possible to perform the double integral using an iterated integration method if R = [a, b] x [c, d].
In this case, according to the iterated integral, the function f(x,y) is integrated with respect to y first, with f(x) being treated as a constant, and then integrated with respect to x, with limits of x being applied and simplified.
Also, in the form of dA = dxdy = dydx, there are two methods to set up the limits for integrals when working with double integrals. You also can solve integration equations with an online iterated integral calculator.
Triple Integration
A triple integral is a form of multiple integrals in calculus that falls within the category of definite integrals. Specifically, it is defined as the multiple integrals of the function with three variables across the area R3 of the coordinate plane.
Whenever there is a function in three variables, f(x,y,z), then the triple integral of the function over the three-dimensional region W is written as
∭w f(x,y,z) dxdydz.
Triple integration calculator is an integral of the form ∭DdV, which implies that we are adding up small amounts of volume along the length of a solid area D in the plane. Moreover, the volume of a solid area in space could be calculated following the triple integrals method.
The differentials of triple integrals may be ordered in six distinct ways, which allows us to define limits for our integrals in six different ways when dealing with triple integrals.
Integration by Parts
To combine the results of two or more functions, the technique of integration by parts is employed. The integration of the two functions f(x) and g(x) has the form f(x) . g(x). The integration by parts technique is the inverse of the product rule of differentiation.
While integrating integrals using this technique, the first function f(x) is chosen in such a manner it has a derivative formula, while the second function g(x) is selected considering it has an integral of that function.
It is calculated as follows: integration of (first function x second function) = (first function) x (Integration of Second Function – Integration of (Differentiation of First Function x Integration of Second Function). While mathematically we can denote it as
∫f(x).g(x).dx = f(x)∫g(x).dx−∫(f′(x)∫g(x).dx).dx+C
By dividing the formula into two parts, integration by parts is carried out such that we can observe in the second part, the derivative of the first function f(x) while the integral of the second function g(x) in both parts after the formula has been divided into two parts.
The letters ‘u’ and ‘v’ are commonly used in integration by parts to denote the functions f(x) and g(x) respectively. Therefore, integration of uv formula using the notation of ‘u’ and “v” is represented by the following equation
∫ u dv = uv – ∫ v du
