Introduction to Integrals
Integrals compute the area under a curve, the reverse of derivatives. They’re essential for finding total quantities like distance or volume.
Definition
The definite integral \( \int_a^b f(x) \, dx \) is the area under \( f(x) \) from \( x = a \) to \( x = b \). The indefinite integral \( \int f(x) \, dx \) is the antiderivative.
Example: \( \int x \, dx \)
The antiderivative of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} + C \]
\( C \) is the constant of integration. For definite integral \( \int_0^2 x \, dx \):
\[ \left[ \frac{x^2}{2} \right]_0^2 = \frac{2^2}{2} - \frac{0^2}{2} = 2 \]
Graphical View
Area under \( y = x \) from 0 to 2:
Rules
Power Rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (if \( n \neq -1 \)). Examples:
- \( \int x^2 \, dx = \frac{x^3}{3} + C \)
- \( \int 1 \, dx = x + C \)
Applications
Integrals calculate distance from velocity, volume of solids, and work done by a force.