Integral is a fundamental type of calculus that allows us to find the area under the curve. The term integral has higher orders such as first, second, third, etc. These are known as sub-types of integral in calculus.
In this post, we are going to explain the order-wise concepts of integral along with rules, formulas, and solved examples.
What is integral?
In the mathematical branch, named calculus, the term integral lies that is useful for finding the new function and area under the curve with respect to the integrating variable. The main purpose of this type of calculus is to find the inverse of integral, volume, displacement, and other mathematical quantities.
The integral of the function is also known as the first integral of the given function. Here are the methods and formulas for finding the new function and area under the curve.
In the first integral, the definite and indefinite integrals are fundamental concepts that are used in different fields. Such as definite integral allows us t find the area, volume, displacement, and many other mathematical quantities.
While the indefinite integral allows us to solve differential equations and understand the behavior of the function.
Methods to Solve Integrals
Below are a few methods to solve the first integral:
An integral calculator by Meracalculator is an alternate way to find the solution of integral problems to avoid the above manual calculations.
What are second and third integrals?
The second and third integrals are also known as double and triple integrals.
The second integral (a.k.a double integral) is the second order of integral that allows us to evaluate the two-dimensional volume under the curve. The function of this order can be represented as f(u, v) and denoted as:
∫∫ f(u, v) du dv
The third integral (a.k.a triple integral) is the third order of integral that allows us to evaluate the volume under the curve in a three-dimensional region. The function of this order can be represented as f(u, v, w) and denoted as:
∫∫∫ f(u, v, w) du dv dw
How to solve double and triple integrals?
Let us take the above examples and find their second and third integrals to understand accurately.
Role of Double & Triple Integrals
|Double integral||Triple Integral|
|It extends the concept of the first integral into 2-D spaces.||It extends the concept of the first integral into 3-D spaces.|
|The volume under the surface in 2D regions can be evaluated with the help of double integral||The volume under the surface in 3D regions can be evaluated with the help of triple integral|
|It is helpful for evaluating the average values of a function over a region||It is essential for calculating mass, center of mass, and moment of inertia for three-dimensional objects.|
We have explored all the basics of calculating the first, double, and triple integral of the given function to deal with the 1, 2, & 3 dimensional regions. You can explore the methods and types of integrals and take assistance from the solved examples.