Symbol Ab Definite Interval Calculator

What is Symbol AB Definite Interval?

The symbol ∫[a,b] represents a definite integral, which calculates the exact area under the curve of a function f(x) between two points a and b on the x-axis. This concept is fundamental in calculus and has applications in physics, engineering, and economics.

Definite integrals provide precise measurements of quantities like distance traveled, work done, or accumulated change. They differ from indefinite integrals, which represent a family of antiderivatives.

How to Calculate Symbol AB Definite Interval

To calculate the definite integral of a function over interval [a, b]:

1. Identify the function f(x) you want to integrate
2. Determine the lower limit a and upper limit b
3. Find the antiderivative F(x) of f(x)
4. Evaluate F(b) - F(a) to get the definite integral value

This process gives you the exact area under the curve between points a and b.

Formula

For common functions, antiderivatives include:

• ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
• ∫e^x dx = e^x + C
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C

Example Calculation

Let's calculate ∫[1,3] x² dx:

1. Find the antiderivative of x²: (x³)/3 + C
2. Evaluate at upper limit (3): (3³)/3 = 9
3. Evaluate at lower limit (1): (1³)/3 = 1/3
4. Subtract: 9 - (1/3) = 26/3 ≈ 8.6667

The area under x² from 1 to 3 is approximately 8.6667 square units.