Lecture #23: The Fundamental Theorem of Calculus

Velocity and Distance

Suppose \(v(t)\) is the velocity of an object of an object at time \(t\) and consider the distance traveled over the time interval \(\Delta t\).

\[ \begin{aligned} \int_a^b v(t) \, dt &= \lim\limits_{n \to \infty} \sum_{i = 1}^n v(t_i) \Delta t \\ &= \textrm{distance traveled (displacement)} \\ &= \textrm{area under the graph of } v(t) \\ &= f(b) - f(a) \end{aligned} \]

The Fundamental Theorem of Calculus

Suppose \(f(t)\) is a continuous function on the interval \([a, b]\) and that \(F(t)\) is an antiderivative of \(f(t)\). Then

\[ \int_a^b f(t) \, dt = F(b) - F(a) \]

The Fundamental Theorem of Calculus

Notes:

This makes evaluating definite integrals easy, provided you can find an antiderivative.

Example: The Fundamental Theorem of Calculus

Example: At 10 a.m. A cistern at a rate of \(R(t) \, \frac{gallons}{hour}\), where \(R(t)\) has the graph given below.

Estimate the amount of water in the tank at noon.

Example: The Fundamental Theorem of Calculus

Examples:

1. Find \(\displaystyle\int_0^2 3x+1 \, dx\).
2. Find \(\displaystyle\int_1^3 (x-1)^2 \, dx\).
3. Find \(\displaystyle\int_0^{\frac{\pi}{2}} \sin(x) \, dx\).
4. Find \(\displaystyle\int_0^3 e^x \, dx\).
5. Find \(\displaystyle\int_0^{\pi} \cos(x) + \dfrac{1}{\sqrt{1 - x^2}} \, dx\).

Some Theorems for Definite Integrals

Theorem: \(\displaystyle\int_a^b c \cdot f(x) + d \cdot g(x) \, dx = c \displaystyle\int_a^b f(x) \, dx + d \displaystyle\int_a^b g(x) \, dx\).

The Fundamental Theorem of Calculus

Every continuous function \(f(x)\) has an antiderivative.

The Fundamental Theorem of Calculus

If \(f(x)\) is a continuous function on an interval and \(a\) is any point, then the function:

\[ F(x) = \int_a^x f(t) \, dt, \]

is an antiderivative of \(f(x)\).