The Fundamental Theorem of Calculus

Engineering Context

MAE: Calculating change in energy of a system as it undergoes a thermodynamic process.

\[ \left( \frac{dE}{dt} \right)_{\text{system}} = \frac{d}{dt} \int e \rho \, dV + \int e \rho \vec{v} \, d\vec{A} \]

ECE: Used in the relationship between current I and electrical charge Q in a capacitor:

\[ Q(t) = Q_0 + \int_{0}^{t} I(\tau) \, d\tau \]

BE: Integration is used in pharmacokinetics, or the study of bioavailability, concentration, and movement of drugs in the bloodstream. For example, the area under the curve (AUC) in pharmacokinetics is the integral of drug concentration in blood plasma over time. AUC is useful in determining drug exposure and average concentration over a certain period of time. It is also a good measure of bioavailability of a drug.

CEE: Calculating volumes of cuts and fills for construction projects.

\[ \iint_R f(x, y) \, dA \]

The Essentials

There are two parts to the Fundamental Theorem of Calculus.

Part 1:

If \( f(t) \) is continuous of over [a,b] and x is in the interval [a,b]:

\[ F(x) = \int_{a}^{x} f(t) \, dt \]
\[ \frac{du}{dt} F(x) = \frac{du}{dt} \int_{a}^{x} f(t) \, dt = f(x) \]
\[ F'(x) = f(x) \]

Part 2:

\[ \int_{a}^{b} f(x) \, dx = F(x) \Big|_{a}^{b} = F(b) - F(a) \]

where \( f(x) \) is continuous over interval [a,b]

A Deeper Dive

Understanding the origin of the Fundamental Theorem of Calculus can help us have a deeper appreciation for it. The Fundamental Theorem of Calculus is a theorem that connects integration and differentiation, and it originated in the 17th century. Typically the credit for discovering calculus goes to Isaac Newton and Gottfried Wilhelm Leibniz, though there were many others who explored the relationship between differentiation and integration.

Even though this theorem was first developed in the late 17th Century, it took nearly two hundred years for the ”Fundamental Theorem of Calculus” to be accepted as its name. In 1898, Daniel A. Murray published a book called An Elementary in the Integral Calculus. His book was the first to use the the word ”fundamental” to describe this theorem. By the the early 1900s, the name became widely accepted.

Since calculus is the study of derivatives and integrals, it only makes sense that this theorem would be termed ”Fundamental”.

Proof

Limit definition of a derivative: \( F'(x) = \lim_{h \to 0} \frac{F(x + h) - F(x)}{h} \)
If \( f(x) \) is the derivative of \( F(x) \), then \( \int f(t) = F(t), \text{ and } F'(x) = \lim_{h \to 0} \frac{1}{h} \left[ \int_{a}^{x+h} f(t) \, dt - \int_{a}^{x} f(t) \, dt \right] \)
\[ F'(x) = \lim_{h \to 0} \frac{1}{h} \left[ \int_{a}^{x+h} f(t) \, dt + \int_{x}^{a} f(t) \, dt \right] \]
\[ F'(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} f(t) \, dt \]

Mean Value Theorem states:

\( \frac{1}{h} \int_{x}^{x+h} f(x) \, dx = f(c) \) , where c is some point between \( x \) and \( x + h \).
\[ F'(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} f(t) \, dt = \lim_{h \to 0} f(c) \]

Since c is between x and x + h, c approaches x as the interval between x and x + h gets smaller and smaller, or in other words as h approaches 0:

\[ \lim_{h \to 0} f(c) = \lim_{c \to x} f(c) = f(x) \]
\( F'(x) = f(x) \)

Practice

Luke and Megan are racing their toy cars on a flat straight track. The race will be 10 seconds long. After 10 seconds whoever’s car has traveled farthest wins the race. Luke’s car has a velocity of \( f(t) = \frac{1}{2t + 2} \text{ ft/sec} \) and Megan’s car has a velocity of \( g(t) = \cos(t) + 5 \text{ ft/sec} \). Who wins the race?

Solution

Since we are given velocity equations, we need to integrate them in order to get position equations. (Note that the bounds for our integral will be from 0 to 10 since we are measuring who travels farthest between 0 and 10 seconds)

Luke:

\[ \int_{0}^{10} \frac{1}{2} t + 2dt = \frac{1}{4} t^2 + 2t \big|_{0}^{10} = (25 + 20) - 0 = 45 \]

Megan:

\[ \int_{0}^{10} \cos(t) + 5 \, dt = \sin(t) + 5t \Big|_{0}^{10} = \sin(10) + 50) - 0 = 49.5 \]

Since Luke traveled 45 ft in 10 seconds and Megan traveled 49.5 ft in 10 seconds, Megan wins the race!

More Resources