Calculus How To

Integrals / Integral Calculus

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The integral, along with the derivative, are the two fundamental building blocks of calculus. Put simply, an integral is an area under a curve; This area can be one of two types: definite or indefinite. Definite integrals give a result (a number that represents the area) as opposed to indefinite integrals, which are represented by formulas. Indefinite Integrals (also called antiderivatives) do not have limits/bounds of integration, while definite integrals do have bounds. Watch the video for a quick introduction on to definite integrals, or read on below for more definitions, how-to articles and videos.

Integrals: Definitions

General How-To Integrals

Integral Calculus Advanced Problem Solving

Integral Kernel

An integral kernel is a given (known) function of two variables that appears in an integral equation; This unknown function appears with an integral symbol.

The kernel is symmetric if If K(x, y) = K(y, x).

Notation for the Integral Kernel

The kernel is denoted by K(x, y):
integral kernel

As well as K(x, y), you might also see slightly different notation depending on what variables are used in the equation. For example:

  • A(x, y),
  • Ta(x, y), or
  • K(x, x′).

What notation is used sometimes depends on exactly what the kernel is representing. Some specific representations include (Wolf, 2013):

  • A translation operation 𝕋a: Ta(x, y),
  • Inversions: I0(x, y),
  • The operator of differentiation: ∇(x, y).

Avramidi (2015) describes an integral operator on the Hilbert space L2 ([a, b]) as follows:
alternate notation

Where the function K(x, x′) is the integral kernel. Note that the author also uses “K” on the left hand side of the equation to denote the operator, a distinction that “…shouldn’t cause any confusion because the meaning of the symbol is usually clear from the context”.

Integral Kernel, or Symbol?

Although the term “integral kernel” is widely used, many authors prefer the alternate term symbol instead, to avoid confusion with many other meanings for the word kernel in mathematics. For example, in geometry, a kernel is the set of points inside a polygon from where the entire boundary of the polygon is visible; In statistics, a kernel is a weighting function used to estimate probability density functions for random variables in kernel density estimation.


Avramidi, I. (2015). Heat Kernel Method and its Applications 1st ed. Birkhäuser
Paulsen, V. & Raghupathi, M. (2016). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces.
Wolf, K. (2013). Integral Transforms in Science and Engineering. Springer Science & Business Media.

How to find the area between two curves in integral calculus

Finding the area between two curves in integral calculus is a simple task if you are familiar with the rules of integration (see indefinite integral rules). The easiest way to solve this problem is to find the area under each curve by integration and then subtract one area from the other to find the difference between them. You may be presented with two main problem types. The first is when the limits of integration are given, and the second is where the limits of integration are not given.

Area Between Two Curves: Limits of Integration Given

Example problem 1: Find the area between the curves y = x and y = x2 between x = 0 and x = 1.

Step 1: Find the definite integral for each equation over the range x = 0 and x = 1, using the usual integration rules to integrate each term. (see: calculating definite integrals).

Step 2: Subtract the difference between the areas under the curves. You’ll need to visualize the curves (sketch or graph the curves if you need to); you’ll want to subtract the bottom curve from the top one. The curve on top here is f(x) = x, so:
1213 = 16.

Limits of Integration NOT Given

Example problem: Find the area between the curves y = x and y = x2.

Step 1: Graph the equations. In most cases, the limits of integration will be clear, especially if you’re using a TI-calculator with an Intersection feature (just find the intersections of the two graphs). If you can find the intersection by graphing, skip to Step 3.

Step 2: Find the common solutions of these two equations if you cannot find the intersection by graphing (treat them as simultaneous equations).
Substituting y = x for x in y = x2 gives an equation y = y2, which has only two solutions, 0 and 1.
Putting the values back into y = x to give the corresponding values of x: x = 0 when y = 0, and x = 1 when y = 1. The two points of intersection are (0,0) and (1,1).

Step 3: Complete the steps in Example Problem 1 (limits of integration given) to complete the calculation.
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Stephanie Glen. "Integrals / Integral Calculus" From Calculus for the rest of us!

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