What Is an Integral?

An integral adds up infinitely many, infinitely small pieces to find the total. It answers questions like, "How much area is under this curve?" or "How far have I traveled if I know my speed at every instant?"

Two Views: Area and Accumulation

• Area: The definite integral of a function from \( a \) to \( b \) finds the area under the curve between \( x = a \) and \( x = b \).
• Accumulation: Integrals can also represent the accumulation of quantities, like total distance from a speed function.

The Fundamental Theorem

The fundamental theorem of calculus connects derivatives and integrals: taking a derivative and then integrating gets you back to where you started!

\[ \int_a^b f(x) , dx = F(b) - F(a) \] where \( F \) is any function whose derivative is \( f \).