Calculus
Study of continuous change through derivatives and integrals.
Integrals and Area Under Curves
What Is an Integral?
Integrals are the opposite of derivatives. They help us combine tiny pieces to find totals, like adding up lots of thin strips to find the area under a curve.
The Notation
Integrals use the symbol \( \int \). The definite integral from \( a \) to \( b \) is written as: \[ \int_a^b f(x),dx \] This measures the area under the curve from \( x = a \) to \( x = b \).
Why Use Integrals?
- To calculate areas, volumes, and totals
- To solve problems involving accumulation (like distance traveled)
- To undo derivatives (finding the original function)
Fundamental Theorem of Calculus
This links derivatives and integrals: integrating a derivative brings you back to the original function!
Examples
The area under \( y = x^2 \) from 0 to 1 is \( \int_0^1 x^2 dx = \frac{1}{3} \).
Finding the total distance a car travels when you know its speed at every instant.
In a Nutshell
Integrals add up infinitely many small pieces to find area, volume, and more.