Multiple Integrals
Multiple integrals are integrals with more than one variable. There are two types of multiple integrals. They are:
- Double Integral
- Triple Integral
Double Integral
The integral of two variable functions under a specified region is called double integration. It is denoted by ∬.
Let’s see the following example to learn more about the process of double integral.
Example: Evaluate the double integral ∬R x2 + y2 dA, where R is the region bounded by the curves y = x2 and y = 2x.
Answer:
The curves y = x2 and y = 2x intersect at the point (0, 0) and (2, 4).
Thus, the limit of integration for variable x is: 0 ≤ x ≤ 2.
and for interval [0, 2], x2 ≤ 2x.
∬R x2 + y2 dA = [Tex]\int_{0}^{2} \int_{x^2}^{2x} (x^2 + y^2) dy dx [/Tex]
Let I’ = [Tex]I’ = \int_{x^2}^{2x} (x^2 + y^2) dy [/Tex]
[Tex]\Rightarrow I’ = \int_{x^2}^{2x} (x^2 + y^2) \, dy \\ \Rightarrow I’ = \left[ x^2y + \frac{y^3}{3} \right]_{x^2}^{2x} \\ \Rightarrow I’ = 2x^3 + \frac{8x^3}{3} – \frac{x^6}{3} + \frac{x^6}{3} \\ \Rightarrow I’ = \frac{14x^3}{3} [/Tex]
Thus, ∬R x2 + y2 dA = [Tex]\int_{0}^{2} \frac{14x^3}{3} dx [/Tex]
⇒ ∬R x2 + y2 dA = [Tex]\frac{14}{3} \times \left[\frac{x^4}{4}\right]_{0}^{2} [/Tex]
⇒ ∬R x2 + y2 dA = (14/3) × [(16/4) – 0]
⇒ ∬R x2 + y2 dA = 56/3
Read More: Double Integral
Triple Integral
The integral of three variable functions under the 3-D region is called triple integration. It is denoted by ∭.
Let’s consider an example to learn how to calculate the triple integral.
Certainly! Here’s the direct solution using LaTeX code for the triple integral example:
Example: Evaluate the triple integral ∭V xyz dV, where V is defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 3.
Solution:
Given the region V: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 3, the triple integral can be evaluated as follows:
[Tex]\begin{aligned} \iiint V xyz \, dV &= \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} xyz \, dz \, dy \, dx \\ \Rightarrow \iiint V xyz &= \int_{0}^{1} \int_{0}^{2} \left[ \frac{xy z^2}{2} \right]_{0}^{3} \, dy \, dx \\ \Rightarrow \iiint V xyz &= \int_{0}^{1} \int_{0}^{2} \frac{27xy}{2} \, dy \, dx \\ \Rightarrow \iiint V xyz &= \int_{0}^{1} \left[ \frac{27xy^2}{4} \right]_{0}^{2} \, dx \\ \Rightarrow \iiint V xyz &= \int_{0}^{1} 27x \, dx \\ \Rightarrow \iiint V xyz &= \left[ \frac{27x^2}{2} \right]_{0}^{1} \\ \Rightarrow \iiint V xyz &= \frac{27}{2} \end{aligned} [/Tex]
So, the value of the triple integral ∭V xyz dV over the region V is 27/2.
Integral Calculus
Integral Calculus is the branch of calculus that deals with topics related to integration. Integrals are major components of calculus and are very useful in solving various problems based on real life. Some of such problems are the Basel problem, the problem of squaring the circle, the Gaussian integral, etc. Integral Calculus is directly related to differential calculus.
This article is a brief introduction to Integral Calculus, including topics such as fundamental theorems of integral calculus, types of integral, and integral calculus formulas, definite and indefinite integrals with their properties, applications of integral calculus, and their examples.
Table of Content
- What is Integral Calculus?
- Fundamental Theorems of Integral Calculus
- Integral Definition
- Types of Integrals
- Definite Integrals
- Definite Integral Formula
- Properties of Definite Integrals
- Indefinite Integrals
- Properties of Indefinite Integrals
- Improper Integrals
- Multiple Integrals
- Integral Calculus Formulas
- Methods to Find Integrals
- Applications of Integral Calculus
- Differential vs Integral Calculus
- Integral Calculus Examples
- Practice Problems on Integral Calculus
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