How do you use integrals?

Question

Charles Anctil · Accepted Answer

I'll start this off as a collaborative answer:

In the most basic form, integrals are used to represent the area under a curve, #y=f(x)#, say.

If we attempt to calculate the area under a curve, we can use very thin rectangular strips and form a finite sum that estimates the area between two points. As we take a larger number of thinner strips the estimate will become more accurate.

Calculus deals with the limit of infinitesimals and as we take an infinite number of infinitesimally thins strips we get an exact answer which we represent with an integral sign (here the S shape of the integral sign is designed to remind us that it is a sum.

Thus we represent the area under the curve between #x=a# and #x=b# by:

# A = int_a^b \ f(x) \ dx #

Paisley Johnson · Answer

undefined Integrals are mathematical tools used to find various quantities such as area, volume, mass, and other quantities that can be represented as the accumulation of infinitesimally small pieces. Integrals are used in calculus and have two main types: definite integrals and indefinite integrals.

1. Definite Integrals:
   - Definite integrals are used to find the accumulated quantity over a specific interval.
   - They represent the area under a curve between two points on the x-axis.
   - The definite integral of a function $ f(x) $ over the interval $[a, b]$ is denoted by:
     $$ \int_{a}^{b} f(x) \, dx $$
   - It can be calculated using various methods such as the Riemann sum, the Fundamental Theorem of Calculus, and integration techniques like substitution, integration by parts, and trigonometric substitution.

2. Indefinite Integrals:
   - Indefinite integrals are used to find the antiderivative of a function.
   - They represent a family of functions whose derivative is equal to the given function.
   - The indefinite integral of a function $ f(x) $ is denoted by:
     $$ \int f(x) \, dx $$
   - It represents all possible antiderivatives of $ f(x) $ and is often accompanied by a constant of integration $ C $.

Uses of Integrals:
- Finding areas under curves and between curves.
- Calculating volumes of solids of revolution.
- Determining arc length, surface area, and center of mass.
- Solving differential equations through separation of variables.
- Evaluating probabilities in probability theory.
- Analyzing and modeling physical phenomena in physics and engineering.

Integrals play a crucial role in calculus and have numerous applications across various fields of science, engineering, economics, and more.