Integration

Introduction to Integrals

Welcome to our exploration of integrals, a fundamental concept in calculus. Integrals are powerful mathematical tools used to determine quantities such as area, volume, and accumulated change. They provide a method for finding the total accumulation of a quantity over an interval, making them indispensable in various fields, including physics, engineering, economics, and more.

In this section, we'll delve into the principles of integration, understand its significance, and explore different techniques for evaluating integrals. Whether you're a student learning calculus for the first time or a seasoned professional looking to refresh your knowledge, join us on this journey through the world of integrals.

Integration Definition

Integration is a fundamental concept in mathematics, particularly in calculus, that involves finding the accumulation of a quantity over an interval. In simpler terms, it deals with determining the total amount or value of something over a given range.

Mathematically, integration is denoted by the symbol ∫ (the integral symbol) and is typically represented as the area under a curve. The process of integration is the reverse operation of differentiation, and it involves finding an antiderivative (also known as the indefinite integral) of a function.

Given a function f(x), the integral of f(x) with respect of x over the integral [a,b]

[a,b] is denoted by:

${\int }_a^{b}f\left(x\right)dx$

This expression represents the accumulation of the function f(x) over the interval [a,b] where dx indicates the differential element with respect to x. Geometrically, this integral represents the area under the curve of f(x) between the limits a and b on the x-axis.

Integration has various applications across mathematics, physics, engineering, economics, and many other fields. It is used to solve problems related to area, volume, probability, optimization, and more.

In summary, integration is the process of finding the total accumulation of a quantity over a given interval, and it plays a crucial role in understanding and solving a wide range of mathematical and real-world problems.

Types of Integral: Definite

Definite integrals are a type of integration that involves finding the accumulation of a function's values over a specific interval. Mathematically, a definite integral represents the area under the curve of a function between two specified limits on the x-axis. It provides a way to calculate the net accumulation or total amount of a quantity over a given range. The result of a definite integral is a single numerical value, representing the total accumulated quantity over the specified interval.

Definite Integral Examples

Say that we have a question,

${\int }_0^1\frac{6x}{{\left(x^2+1\right)}^3}dx$

Here are the steps you need to follow,

• Find a suitable substituent, usually has an exponent. In this case, it's
  ${\left(x^2+1\right)}^3$
• Assume that the substituent is the variable 'u'
  ${\left(u\right)}^3$
  
  ${\int }_0^1\frac{6x}{{\left(u\right)}^3}dx$
• Start substituting
  $\frac{du}{dx}=2x$
  
  $du=2xdx$
  
  ${\int }_0^1\frac{6x}{{\left(u\right)}^3}dx$
  
  ${\int }_0^1\frac{3du}{{\left(u\right)}^3}$
• Move the constant number infront of the integral
  $3{\int }_0^1\frac{du}{{\left(u\right)}^3}$
• Next, use antiderivative
  $3\left(\frac{1}{3+1}{u}^{3+1}\right)$
• Substitute the value of 'u', with the boundaries
  $3\left(\frac{1}{4}{1}^4\right)-3\left(\frac{1}{4}{0}^4\right)$
• Calculate the result
  $\frac{3}{4}-0=\frac{3}{4}$

Types of Integral: Indefinite

Indefinite integrals, also known as antiderivatives, are the reverse operation of differentiation. They involve finding a function whose derivative is equal to a given function. In other words, an indefinite integral represents a family of functions that differ by a constant. When computing indefinite integrals, the result includes a constant of integration (often denoted as "+C") to account for the infinite number of possible functions that could satisfy the derivative relationship. Indefinite integrals are useful for finding general solutions to differential equations and for evaluating integrals symbolically without specific limits.

Indefinite Integral Examples

Solve:
${\int }_{}^{}\left(4x^3+8x^2-9\right)dx$

• Apply the integral to each term separately
  ${\int }_{}^{}4x^{3}dx+{\int }_{}^{}8x^{2}dx-{\int }_{}^{}9dx$
• Find the antiderivative of each term
  $4\left(\frac{x^{3+1}}{3+1}\right)+8\left(\frac{x^{2+1}}{2+1}\right)-9\left(\frac{x^{0+1}}{0+1}\right)$
  
  $4\left(\frac{x^4}{4}\right)+8\left(\frac{x^3}{3}\right)-9\left(\frac{x^1}{1}\right)$
• Simplify
  $x^4+\frac{8x^3}{3}-9x$
• Since it is indefinite integral, we need to add constant C at the end.
  $x^4+\frac{8x^3}{3}-9x+C$

Solve:
$\underset{}{\overset{}{\int }}\sqrt[3]{x^4}dx$

• Rewrite the exponents into fractional form
  $\underset{}{\overset{}{\int }}{x}^{\frac{4}{3}}dx$
• Next, use antiderivative
  $\left(\frac{x^{\frac{4}{3}+1}}{\frac{4}{3}+1}\right)$
• Simplify
  $\left(\frac{x^{\frac{7}{3}}}{\frac{7}{3}}\right)$
  
  $\frac{3}{7}{x}^{\frac{7}{3}}$
• Since it is indefinite, don't forget to add constant C at the end.
  $\frac{3}{7}{x}^{\frac{7}{3}}+C$

The Area Above a Curve

If f(x) is negative and continuous on the interval a ≤ x ≤ b, then the area bounded by y = f(x), the x-axis, and the vertical lines x = a and x = b is given by

$A=-{\int }_a^{b}f\left(x\right)dx$ or $-{\int }_a^{b}ydx$

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For $y=f\left(x\right)>0$, area = ${\int }_a^{b}f\left(x\right)dx$

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For $y=f\left(x\right)<0$, area = $-{\int }_a^{b}f\left(x\right)dx$

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For $x=f^{-1}\left(y\right)>0$, area = ${\int }_c^d{f}^{-1}\left(y\right)dy$

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For $x=f^{-1}\left(y\right)<0$, area = $-{\int }_c^d{f}^{-1}\left(y\right)dy$