integrand

Imagine you're building with LEGO bricks. The whole structure is like an integral: a total made from many tiny pieces. The part that gives that total its shape is like the function under the integral sign. In calculus, this is called the integrand. It is the function, or more generally the expression, that you integrate. It is the part inside the long, stretched-out "S" symbol (∫), and it is what gets added up in very small slices when you find an area under a curve or work out another quantity. For example, if you want the area under the curve for $f(x)=x^2$, then $x^2$ is the integrand.

The word "integrand" comes from calculus and refers to the function, or more generally the expression, that is being integrated. Think of integration as a way of adding up an enormous number of very small pieces to find a total quantity, such as the area under a curve, the volume of a solid, or the probability linked to a density function.

The integrand is the part you are adding up. It appears under the integral symbol (∫). In a simple example like $\int x^2 \, dx$, the main integrand is $x^2$. More broadly, people also use the word for the whole expression under the sign, depending on context. The "$dx$" shows that the integration is being done with respect to $x$.

You won't typically use "integrand" in everyday conversation. It's a technical term used by mathematicians, physicists, engineers, and anyone working with higher-level math. For example, a professor might say, "The integrand in this definite integral represents the probability density function," or, "Choosing an appropriate substitution can simplify the integrand and make the integration process easier."

Understanding the concept of an integrand is important for learning integration well. It is not only about memorizing formulas. It is about seeing what quantity is being added up and how that expression behaves. So, the next time you see the ∫ symbol, look at what sits beneath it: that is the part the calculation is working on.