Calculus

Study of continuous change through derivatives and integrals.

Limits and ContinuityDerivatives and Rates of ChangeIntegrals and Area Under Curves
Applications of DerivativesTechniques of IntegrationInfinite Series and Convergence
Modeling MotionEconomics and OptimizationBiology and Population Growth
Breaking Down ProblemsMemorizing Key FormulasChecking Your Work
Limits and Continuit...Derivatives and Rate...Integrals and Area U...Applications of Deri...Techniques of Integr...Infinite Series and ...Modeling MotionEconomics and Optimi...Biology and Populati...Breaking Down Proble...Memorizing Key Formu...Checking Your Work
Basic Concepts

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.

Key Terms

Integral
A way to sum up small pieces to find a total, like area under a curve.
Antiderivative
A function whose derivative is the original function.
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Applications of Derivatives