JUMP TO TOPIC

Calculus, the mathematical study of change and motion, often brings two closely related but distinct concepts to the fore: **antiderivatives and integrals**. While these terms are occasionally used interchangeably in casual conversations, they address different aspects of calculus.

Let’s embark on a journey to understand the nuances between them and explore their significance in the realm of mathematics.

**What is the Difference Between an Antiderivative and an Integral **

The main difference between **antiderivative** and** integral** is that the **antiderivative** of a function is essentially a reverse operation to differentiation; it seeks a function whose derivative is the given function.

On the other hand, the **integral** of a function primarily concerns itself with the area under a function’s curve. While the indefinite **integral** can be synonymous with the **antiderivative**, presenting a family of functions as its outcome, the definite **integral** provides a distinct value, representing the signed area between the function and the x-axis over a specific interval.

In essence, while both concepts operate within the realm of calculus, they serve different mathematical purposes and insights.

### What is an Antiderivative?

**Definition**

The **antiderivative** of a function $f(x)$ is another function $F(x)$ such that the derivative of $F(x)$ is $f(x)$. In other words, if $F_{′}(x)=f(x)$, then $F(x)$ is an** antiderivative** of $f(x)$.

**Notation**

While there’s no standard notation specifically for **antiderivatives**, they’re sometimes represented as $∫f(x)dx$, without limits of integration.

**Uniqueness **

One key aspect to note about **antiderivatives** is that they aren’t unique. If $F(x)$ is an **antiderivative** of $f(x)$, then $F(x)+C$, where $C$ is any constant, is also an **antiderivative** of $f(x)$.

For instance, consider the function $f(x)=2x$. Its **antiderivative** could be $x^2$, but it could also be $x^2$ $+5$, $x^2$ $−7$, or $x^2$ $+C$, where $C$ is any real number.

### What is an Integral?

The **integral** of a function represents the area under the curve of that function over a certain interval. There are two primary types of** integrals** in calculus: definite and indefinite.

**Indefinite Integral**

**Definition**

The indefinite** integral** is essentially the **antiderivative** of a function. It represents the family of functions whose derivative is the given function.

**Notation**

It’s denoted as $∫f(x)dx$. Note that it doesn’t have upper and lower limits.

**Example**

The indefinite** integral** of $f(x)=x$ is $∫xdx= $x^2$ / 2 +C$, where $C$ is the constant of integration.

Figure-1.

**Definite Integral**

**Definition**

The definite **integral** computes the signed area between the function and the x-axis over a specified interval [a, b].

**Notation**

It’s denoted with upper and lower limits: $ f(x)dx, from a to b$.

**Example**

For $f(x)=x$ over the interval [0,2], the definite **integral** is $∫xdx$. for [0,2] This calculates the area under the curve from 0 to 2, which results in a value of 2.

## Distinguishing Between the Antiderivative and Integral

**Purpose****Antiderivative**: Finds a function whose derivative is the given function.**Integral**: Computes areas under curves (definite) or represents the family of functions whose derivative is the given function (indefinite).

**Uniqueness****Antiderivative**: Not unique due to the integration constant.**Definite Integral**: Produces a unique real number since it computes a specific area.**Indefinite Integral**: Not unique for the same reasons as the**antiderivative**.

**Physical Significance****Antiderivative**: Often used in physics to find quantities when rates of change are known, such as determining position from velocity.**Integral**: Used to calculate quantities like area, volume, work, etc.

**Exercise **

### Example 1

Function: $f(x)= $x^2$$

### Solution

**Antiderivative**:

$∫f(x)dx=$x^2$dx= $x^3$ / 3 +C$

where $C$ is an integration constant.

**Definite Integral** (over the interval [0,2]):

$$x^2$dx=$x^3$ / 3 ], form 0 to 2$

$∫$x^2$ dx=8 / 3 −0=8 / 3 $

### Example 2

**Function**: $f(x)∫$e^x$dx$

Figure-2.

### Solution

**Antiderivative**:

$∫f(x)dx=$e^x$dx$e^x$+C$

**Definite Integral** (over the interval [0,1]):

$f(x)dx =$e^x$dx, from 0 to 1$

$∫()=$$$e^x$0 to x=1$

$∫f(xdx$e^{-1}$$

### Example 3:

**Function**: $f(x)=sin(x)$

Figure-3.

### Solution

**Antiderivative**:

$∫f(x)dx=∫sin(x)dx=−cos(x)+C$

**Definite Integral** (over the interval [0,π]):

$∫sin(x)dx=[−cos(x)_{0 to x=}$

$∫sin(x)dx= 2$

## Real-World Implications:

The profound concepts of **antiderivatives and integrals** aren’t confined within the boundaries of academic mathematics; they have extensive applications in real-world scenarios. Understanding their real-world implications provides a richer appreciation for these mathematical tools.

**Physics****Antiderivative**: In kinematics, if the velocity of an object (rate of change of position) is known,**antiderivatives**are used to determine its position or the path it has traveled over time.**Integral**: When calculating work done by a variable force over a specific distance or the electric charge stored over a changing voltage, definite**integrals**come into play.

**Engineering****Antiderivative**: Engineers often resort to**antiderivatives**when they need to reverse engineer rates of change, like determining displacement from a known stress-strain curve.**Integral**: In civil engineering,**integrals**are used to compute the area and volume essential for design, especially in complex structures like bridges and dams. Similarly, in electrical engineering,**integrals**help in analyzing changing currents and voltages over time.

**Economics****Antiderivative**: If an economist knows the marginal cost (the cost to produce one additional item), they can use the**antiderivative**to find the total cost function.**Integral**: Economists use definite**integrals**to calculate quantities like consumer and producer surpluses, which provide insights into market efficiency and welfare.

**Biology and Medicine****Antiderivative**: Used in pharmacokinetics, the study of how drugs are absorbed, distributed, metabolized, and excreted in the body. Knowing the rate at which a drug leaves the bloodstream can help determine the drug’s concentration at different times.**Integral**: In modeling biological processes,**integrals**assist in computations like determining the total number of cells in a growing tumor over time or understanding how blood volume changes in response to certain conditions.

**Environmental Science****Antiderivative**: Useful in predicting how pollutants disperse in natural waters over time, based on their rate of spread.**Integral**: When estimating the total amount of materials required to restore a habitat or the total energy received by a surface during the day,**integrals**provide the necessary computational tools.

**Astronomy**

**Antiderivative**: Helps in determining the orbits of celestial bodies based on their speed and direction.**Integral**: Used to compute the total luminosity of stars or the total mass of galaxies within certain radii.

*All images were created with GeoGebra.*