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Intro to Integrals: Integral Meaning, Definition and Function

Intro to Integrals: Integral Meaning, Definition and Function

Oh boy. Integrals. Calculus wasn’t complicated enough… now we have these big swirly lines, too?!

If that’s what’s running through your head right now, you can stop those thoughts in their tracks. Integral calculus actually isn’t as intimidating as it seems!

Once we really get into it, you might even think integrals are — dare we say — cool?

There’s only one way to find out! Let’s get started.

What is an integral?

Depending on where you are in your learning journey, an integral can represent the answer to a few different questions. At their core in calculus, integration helps you find the anti-derivative of a function; in other words, finding an integral is the inverse of finding a derivative.

As you work through your calculus lessons, you’ll see that there are different types of integrals, including:

The process of finding an integral is called “integration,” so an integral can also be thought of as the product of integration. There are several ways to evaluate integrals, such as the substitution method or partial integration.

Because integration is the inverse of differentiation, integrals are easier to understand when your knowledge of derivatives is already solid. If you still need a little help putting the pieces together, we can help you with derivatives first!

Definition of an integral

As we mentioned, there are a few different types of integrals. That means we’ll have a few different definitions, as well.

Here are the integral definitions that you’ll likely need (and use!) the most:

Integral type What is it? What can it look like?
Definite integral $$\text{A specific area bound by the graph of a function, the } x\text{-axis, and the vertical lines } x = a \text{ and } x = b$$ $$\int_a^b f(x)$$
Indefinite integral All the anti-derivatives of a function $${\int f(x) dx = F(x) + C}$$
Improper integral $$\text{If } f \text{ is continuous on } [a,b\rangle \text{ and discontinuous in } b\text{, then the integral of } f \text{ over } [a,b\rangle \text{ is improper}$$ $$\int_a^{b}f(x)dx=\lim_{c\to {b}^{-}}\int_a^{c}f(x)dx$$
Iterated integral Can represent the volume of a solid in 3D $$\iint\limits_{R}{f(x,y)~dA}=\int_{a}^{b}\!\!\!\int_{c}^{d}{f(x,y)~dy~dx}=\int_{c}^{d}\!\!\!\int_{a}^{b}{f(x,y)~dx~dy}$$

Integral of a function

Because we’re focusing on integral calculus, finding the integral of a function is really at the core of our discussion. It’s the basis for all those different types of integrals and methods we mentioned earlier.

Depending on the problem, you can find the integral of a function using several different methods:

Got a specific problem in front of you? Scan it with your Photomath app to learn how to find the integral in that particular scenario.

The meaning of integrals

We’ve gone through all the definitions and background, but what does all of that really mean? Again, it depends on the context of the problem, but an integral can tell you:

  • The area under a curve on a graph
  • The area between a portion of a function and the $$x$$-axis
  • The volume of water in a bathtub based on the rate of flow from the faucet
  • The center of mass of a vehicle so that its safety features can be fine-tuned
  • The best way to create a 3D model

The more you think about it, the more you’ll start to see all the interesting and important uses of integrals all around you!

Integral notation

Integrals have a basic structure of $${\int f(x) dx}$$, so let’s look at the example $$\int\frac{(3x – 2)^{2}}{x^{3}}dx$$.

The ∫ signals that we’re dealing with an integral. That symbol begins framing the expression, kind of like the start of a set of parentheses.

The expression whose integral we need — called the “integrand” — is a function $$f(x)$$. In the example above, the integrand is $$\frac{(3x – 2)^{2}}{x^{3}}$$.

The $$dx$$ stands for “differential” and can be thought of as the other parenthesis, framing the expression on the right side.

That example doesn’t include limits of integration, which means the above is an indefinite integral.

A definite integral includes limits of integration, which you’ll notice at the top and bottom of the integral symbol. Here’s the structure, along with a populated example:

$$\int_a^b f(x)dx$$

The $$a$$ and $$b$$ (or $$-1$$ and $$1$$) are the limits of integration that define the interval to which we’re confined.

In mathematical terms, we would describe a definite integral as “the integral of the function $$f(x)$$ with respect to the variable $$x$$, on an interval $$[a, b]$$.”

If you just look at those mathematical descriptions or expressions all at once, it can be a bit overwhelming. Isn’t it easier when you look at each piece individually?

Integral sign

That squiggly, swirly, curly line is the hallmark of an integral, so when you see ∫ on the page, you know you’re dealing with an integral!

Fun fact: The shape of the integral sign is actually an elongated “S” standing for “sum” (it’s the Roman ∫ instead of the Greek ∑). The “S” for “sum” is based on the idea of adding the area of slices under a curve — the more slices you divide the entire area under a curve into, the more precise sum you get. An integral is most precise because the ‘’columns’’ of slices become infinitesimally thin. So cool!

Just need a quick copy-and-paste of the integral sign itself? Here you go: ∫

You can also use these keyboard shortcuts!

  • iOS: [Option] + [B]
  • Windows: [Alt] + 8747

Integral of dx

In the structure of an integral, you’ll see ∫, followed by the integrand, and then “$$dx$$” as a sort of period at the end of the sentence, like this:

$${\int f(x) dx}$$

That $$dx$$ — known as the differential — tells us that $$x$$ is the variable of integration.

On a graph: Integrals as the area under a curve

Want to see what a definite integral can represent in the coordinate system?

Let’s take the example $$\int_1^2(2x+1)dx$$. That integral has the function $$f(x)=2x+1$$ as its integrand, which is a simple linear function. That means its graph is just a straight line!

As we can see from the integral itself, the limits of integration are $$1$$ and $$2$$, meaning our bounds are straight lines parallel to the $$y$$-axis: lines $$x=1$$ and $$x=2$$.

If we draw all that up in the coordinate system, we’ll get something we call an “area under the curve” that is a visual representation of the integral $$\int_1^2(2x+1)dx$$:

  • Graph of an integral

Practice problems

We know, you probably have enough homework to do already. But, if you want to try your skills in an un-graded space — or if you just want to get some extra practice in — we’ve compiled some sample problems to help you build your skills (and confidence!).

  1. $$\int_{\frac{1}{4}}^1(6x^2-\ln x)dx$$
  2. $$\int_{-1}^{1}\frac{x^{2}}{2}dx$$
  3. $$\int(\cos(t^3)+\frac{1}{t})dt$$
  4. $$\int(e^x-e^{2x})dx$$
  5. $$\int\sqrt{2x+6}dx$$

Getting stuck? That’s okay! Take a deep breath and use your Photomath app to scan the problem that’s giving you trouble. We’ll take you through each step at your own pace, in as much detail as you need. Don’t ever forget: you’re not alone!

Here’s how we solve the first practice problem in the app:

What is the rule for integrals?

There are many different rules and properties for integrals, including:

Constant multiple property of integrals $$\int{(c\times f(x))}dx=c\times \int{f(x)}dx$$
Sum rule for integrals $$\int{(f(x) + g(x))}dx=\int{f(x)}dx + \int{g(x)}dx$$
Difference rule for integrals $$\int{(f(x) - g(x))}dx=\int{f(x)}dx - \int{g(x)}dx$$
Substitution rule $$\int{f(\varphi(t))}\varphi^{\prime}(t)dt=\int{f(x)}dx$$
Integration by parts $$\int{u}dv=uv-\int{v}du$$

What are the two types of integrals?

There are several different kinds of integrals, but the two main types are definite and indefinite integrals.

Why are integrals used?

Integrals are used to find the anti-derivative, or the inverse of the derivative. This unlocks many different pieces of information, like area, volume, velocity, and more.