Integrals

Updated

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

Watch the video for an introduction to integrals in Desmos, and open the integral example graph to explore. Then, read on to learn more.

Definite Integrals

GIF that shows typing int in the Graphing Calculator. This automatically changes int into the integrand symbol with an upper and lower bound.

Type int (or integral if working in the Geometry Tool) in an expression line. Then, enter a lower bound, upper bound, integrand, and differential (such a \(dx\)).

f of x is defined as one-tenth x squared plus one. The definite integral from zero to three of f of x evaluates to 3.9.

Take the function \(f(x)=\frac{1}{10}x^2+1\) as an example. If you take the integral from \(0\) to \(3\) of \(f(x)dx\), the definite integral will evaluate to \(3.9\).

f of x y is defined as xy^2 and is graphed in the 3D cube. Expression line 2 evaluates the double integral from 0 to 1 and from 0 to 2 of f(x,y) dx dy which equals 2/3.

In the 3D Calculator, you can take double integrals. For example, if you have a function \(f(x,y) = xy^2\), you can take 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 \(1\).

 

Indefinite Integrals and Infinite Limits of Integration

The integral from zero to x of t squared dt is graphed in the graphing calculator.

You can also 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 \(x\).

f of x is defined as one over x squared and graphed. The integral from one to infinity of f of x evaluates to one.

Desmos will evaluate convergent integrals with infinite limits. Enter 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 \(1\).

f of x is defined as one over x to the a. a has a slider set in increments of one, running from 1 to 10. As a progresses, the integral from one to infinity of f of x evaluates to undefined, one, then one-half, then one-third, then one-fourth, and so on, all the way to one-ninth.

Divergent integrals will show as undefined. For example, if \(f(x)=\frac{1}{x^a}\), and the slider for \(a\) is set from \(1\) to \(10\), the integral will diverge when \(a=1\) and will show as undefined.

 

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