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Section6.7The Fundamental Theorem of Calculus

When we introduced the definite integral, we also learned about accumulation functions. An accumulation function is a function \(A\) defined as a definite integral from a fixed lower limit \(a\) to a variable upper limit where the integrand is a given rate function \(f\), \begin{equation*}A(x) = \int_a^x f(z)\, dz.\end{equation*} The motivation for such a function was that the definite integral computes the total change of a quantity when the rate of change is given.

The Fundamental Theorem of Calculus provides our guarantee that this makes valid mathematical sense. In particular, we will learn that the derivative of an accumulation function is the same rate function that was used to define the accumulation. Then knowing this fact will allow us to compute any definite integral for which we know an antiderivative of the integrand.

Proof

Most calculus textbooks test understanding of this concept with problems relating to differentiation, including the use of the chain rule.

Example6.7.2

Compute the following derivatives.

  1. \(\displaystyle \frac{d}{dx} \int_1^x e^{-z^2} \, dz\)
  2. \(\displaystyle \frac{d}{dx} \int_0^{\sqrt{x}} \frac{1}{\sqrt{z^4+1}} \, dz\)
  3. \(\displaystyle \frac{d}{dx} \int_{x}^{x^2} \sin(\cos(z))\, dz\)
Solution

Conceptually, it is more important to recognize that definite integrals represent evaluation of accumulation functions and know that those accumulation functions are antiderivatives of the integrand. This leads to the second half of the Fundamental Theorem of Calculus, the evaluation of definite integrals using any known antiderivative.

Proof

The evaluation of definite integrals using the change in an antiderivative occurs so frequently that a notation has been adopted to represent this process. Given any function \(F(x)\), the notation \(\big[ F(x) \big]_{a}^{b}\) means evaluate the change of the expression when \(x\) goes from \(a\) to \(b\): \begin{equation*}\big[ F(x) \big]_{a}^{b} = F(b) - F(a).\end{equation*} When the formula for \(F(x)\) is simple, the brackets can be dropped and replaced by a vertical bar on the right, \begin{equation*} F(x) \big|_a^b = F(b) - F(a).\end{equation*}

Applications of the Fundamental Theorem of Calculus to evaluate a definite integral generally involve two steps. First, identify an antiderivative. Second, evaluate the change in that antiderivative. The evaluation notation described above allows us to represent these two steps when we acknowledge we are using this theorem.

Example6.7.4

Evaluate \(\displaystyle \int_1^4 x^2 \, dx\).

Solution

Evaluation of definite integrals involves recognizing antiderivatives and then evaluating their change.

Example6.7.5

\begin{align*} \displaystyle \int_0^\pi \sin(x) \, dx & \overset{{\mathrm{FTC}}}{=} \big[-\cos(x)\big]_{0}^{\pi}\\ &= -\cos(\pi) - -\cos(0) \\ &= -(-1) - -(1) = 2 \end{align*}

Example6.7.6

\begin{align*} \displaystyle \int_1^2 e^{3x} \, dx & \overset{{\mathrm{FTC}}}{=} \big[\frac{1}{3}e^{3x}\big]_{1}^{2}\\ &= \frac{1}{3}e^{6} - \frac{1}{3}e^{3} \\ &= \frac{e^6-e^3}{3} \end{align*}