Integrals

Use the Desmos Graphing Calculator, Geometry Tool, or 3D Calculator to investigate the beautiful world of integral calculus.

Get started with the video on the right, then dive deeper with the resources and challenges below.

If you'd like to explore the integral graph shown in the video (and open the "visual" folder), click here.

Type int (or integral if working in the Geometry tool) in an expression line to bring up an integration template where you can type in a lower bound, upper bound, integrand, and differential (such as \(dx\)).

Let’s look at an example. If \(f(x)=\frac{1}{10}x^2+1\), and you take the integral from \(0\) to \(3\) of \(f(x)dx\), you’ll see this definite integral evaluates to \(3.9\). By clicking on the images, you can open the graphs and explore.

In Desmos 3D you can take double integrals. For example, try setting \(f(x,y) = xy^2\) and taking the double integral of \(f(x,y)dxdy\) where the bounds for \(x\) are from \(0\) to \(2\) and the bounds for \(y are from \(0\) to \(1\).

It's also possible to graph the output of some indefinite integrals. One way to do so is by including \(x\) in the upper bound, \(0\) in the lower bound, and integrating with respect to a variable other than \(x\). Try graphing the integral from \(0\) to \(x\) of \(t^2 dt\).

Desmos will evaluate convergent integrals with infinite limits. Type infinity or infty into either the upper or lower bound! For example, if \(f(x)=\frac{1}{x^2}\), then the integral from \(1\) to infinity of \(f(x)dx\) is \(1\).

Divergent integrals will show undefined. Let’s set \(f(x)=\frac{1}{x^a}\) with a slider for \(a\) from \(1\) to \(10\). When \(a=1\), the integral diverges and shows undefined. Click on the GIF and play the slider to see how the graph of the function and evaluation of the integral changes.