Fundamental Theorem of Calculus (Cambridge International A-Level Mathematics)

Fundamental theorem of calculus

What is the fundamental theorem of calculus?

• The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation

  • This form of the Theorem relates to Indefinite Integration
  • An alternative version of the Fundamental Theorem of Calculus involves Definite Integration

What is a constant of integration?

• When differentiating y, constant terms ‘disappear’

  • for constants y = c, $\frac{dy}{dx}=0$
  • graphs of constants are horizontal lines and so have gradient $( \frac{d y}{d x} )$ of 0

• Integrating $\frac{dy}{dx}$, to get y, cannot determine the constant

  • To acknowledge this constant, “+ c” is used
  • c is called the constant of integration

What notation is used in integration?

• integral is the sign for integration

• If it has more than one term the function to be integrated (called the integrand) should be in brackets

  • “Integrate” -– “all of (…)” -– “with respect to x”

• dx means integrate with respect to x, any other letter is treated like a number (ie like a constant)

Worked Example