CLEP Calculus

CLEP Calculus offers students the opportunity to demonstrate their understanding of calculus concepts and applications, enabling them to earn college credit.

Advanced Topics

Integrals and Area Under Curves

Understanding Integrals

Integrals let us add up infinitely many tiny pieces to find totals—like the total area under a curve, or the total distance traveled.

Definite vs. Indefinite Integrals

  • Definite integral: Calculates the exact area under a curve between two points.
  • Indefinite integral: Finds the general formula for all possible areas (includes a "+ C" constant).

\[ \int_{a}^{b} f(x) , dx \]

The Fundamental Theorem of Calculus

This connects derivatives and integrals, showing they’re opposites. If \( F'(x) = f(x) \), then:

\[ \int_{a}^{b} f(x) , dx = F(b) - F(a) \]

Real-World Uses

Integrals measure things like areas, volumes, and total accumulated change.

Key Formula

\[\int_{a}^{b} f(x),dx\]

Examples

  • Finding the area under \( y = x^2 \) between \( x = 0 \) and \( x = 2 \) with \( \int_{0}^{2} x^2 , dx \).

  • Calculating the total distance a car travels using its velocity function.

In a Nutshell

Integrals add up tiny pieces to find totals, like area and accumulated change.

Key Terms

Integral
A mathematical tool for finding the total accumulation of a quantity.
Definite Integral
An integral with upper and lower limits, giving a specific value.
Indefinite Integral
An integral without limits, representing a family of functions.