Lecture #2: Review: Riemann Sums, Definite Integrals & the Fundamental Theorem of Calculus

Riemann Sums

Consider a function \(f(x)\) defined on an interval \(a \le t \le b\), and, given a whole number \(n\), divide the interval into \(n\) subintervals of length \(\Delta t = \frac{(b-a)}{n}\) with endpoints \(t_0, t_1, t_2, \dots, , t_n\).

The left (regular) Riemann sum is:

\[ f(t_0) \Delta t + f(t_1) \Delta t + \dots + f(t_{n-1}) \Delta t = \sum_{i=0}^{n-1} f(t_i) \Delta t. \]

Riemann Sums

Consider a function \(f(x)\) defined on an interval \(a \le t \le b\), and, given a whole number \(n\), divide the interval into \(n\) subintervals of length \(\Delta t = \frac{(b-a)}{n}\) with endpoints \(t_0, t_1, t_2, \dots, , t_n\).

The right (regular) Riemann sum is:

\[ f(t_1) \Delta t + f(t_2) \Delta t + \dots + f(t_n) \Delta t = \sum_{i=1}^{n} f(t_i) \Delta t. \]

Definite Integral

Given a continuous function \(f(x)\) on an interval \(a \le t \le b\), the definite integral of \(f(x)\) over \([a,b]\) is the limit:

\[ \begin{aligned} \int_a^b f(t)\,dt &= \lim\limits_{n \to \infty} \sum_{i=1}^{n} f(t_i) \Delta t, & &\text{or} \\ \int_a^b f(t)\,dt &= \lim\limits_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) \Delta t \end{aligned} \]

provided these limits exist. \(f(x)\) is called the integrand and \(a\) and \(b\) are called the limits of integration.

Example: Definite Integral

Example: Consider the function \(f(x) = x^2 + 1\) over the interval \([0, 2]\). For a given \(n\), \(\Delta t = \frac{2}{n}\) and \(t_i = \frac{2i}{n}\), so that:

\[ \int_0^2 x^2 + 1 \, dx = \lim\limits_{n \to \infty} ((\frac{2i}{n})^2 + 1) \times (\frac{2}{n}). \]

This sum can be thought of as a sum of areas of rectangles the graph of \(f(x)\).

Example: \(n=5\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{4} ((\frac{2i}{5})^2 + 1) \times (\frac{2}{5}) = 4.384 \]

Example: \(n=10\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{9} ((\frac{2i}{10})^2 + 1) \times (\frac{2}{10}) = 4.448 \]

Example: \(n=20\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{19} ((\frac{2i}{20})^2 + 1) \times (\frac{2}{20}) = 4.571 \]

Example: \(n=50\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{49} ((\frac{2i}{50})^2 + 1) \times (\frac{2}{50}) = 4.627264 \]

Example: \(n=100\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{99} ((\frac{2i}{100})^2 + 1) \times (\frac{2}{100}) = 4.646808 \]

Example: \(n=1000\)

\[ \int_0^2 x^2 + 1 \, dx = \sum_{i=0}^{999} ((\frac{2i}{1000})^2 + 1) \times (\frac{2}{1000}) = 4.664668 \]

Example: \(n \to \infty\)

As \(n \to \infty\), this sum becomes the area under the graph of \(f(x)\) over \([a,b]\).

\[ \int_0^2 x^2 + 1 \, dx = \lim\limits_{i \to \infty} \sum_{i=0}^{n-1} ((\frac{2i}{n})^2 + 1) \times (\frac{2}{n}) = \frac{14}{3} \]

The Definite Integral

Estimates of \(\displaystyle\int_a^b f(x)\,dx\) can be obtained with left sums, right sums, sums from midpoint values of \(f(x)\) or more general types of sums.

Example: Definite Integral

Example: Evaluate \(\displaystyle\int_0^2 -2x + 1\,dx\).

\(\displaystyle\int_a^b f(x)\,dx\) is the signed area – that is, the area between the graph of \(f(x)\) and the \(x\)-axis.

Velocity and Distance

Recall the definition of rate of change: \(r = \dfrac{d}{t}\). This implies that \(d = rt\). Suppose \(v(t)\) is the velocity of an object of an object at time \(t\) and consider the distance traveled over the time interval \(\Delta t\).

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

Example: Velocity and Distance

The velocity of an object moving along an axis is described by the picture below. Find the displacement of the object from time \(t=0\) to time \(t=4\).

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.
• Alternatively, \(\displaystyle\int_a^b f'(t) \, dt = f(b) - f(a)\).
• Also, \(F(b) = F(a) + \displaystyle\int_a^b f(t) \, dt\).

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 \(\displaystyle\int_1^3 (x-1)^2 \, dx\).

Example: The Fundamental Theorem of Calculus

Examples:

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

• \(\displaystyle\int_a^b c \cdot f(x) + d \cdot g(x) \, dx = c \displaystyle\int_a^b f(x) \, dx + d \displaystyle\int_a^b g(x) \, dx\).
• If \(a < c < b\), \(\displaystyle\int_a^b f(x) \, dx = \displaystyle\int_a^c f(x) \, dx + \displaystyle\int_c^b f(x) \, dx\).

Note: There is no product or quotient rule for definite integrals.

The Fundamental Theorem of Calculus

Every continuous function \(f(x)\) has an antiderivative. 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 \(f(x)\). In other words:

\[ \frac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x). \]