Integrals

Use the Desmos Graphing 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 graph shown in the video (including taking a look at what's inside the "visual" folder), click here.

Definite Integrals

Calculator keypad function menu opened.  Miscellaneous functions and integral called out. Screenshot.

Simply type int in an expression line to bring up an integration template. Additionally, you can access the integration template from the Functions menu on the keyboard, under Miscellaneous functions.

f of x is defined as one-tenth x squared plus one. The definite integral from zero to four of f of x evaluates to six point three repeating. Screenshot.

Type in your upper bound, lower bound, integrand, and differential (\(dx\) in the example pictured above), and Desmos will evaluate your definite integral.

f of x is defined as one-tenth x squared plus one. The definite integral from a to b has sliders for a and be that run from negative ten to ten. Currently the sliders are set at zero an five.  The definite integral evaluates to nine point one six repeating. Screenshot.

Use variables and sliders in the place of your upper and lower bound to show how the result changes as the bounds change.

 

Indefinite Integrals and Infinite Limits of Integration

The integral from zero to x of t squared. Screenshot.

It's also possible to graph the output of some indefinite integrals 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. The integral from one to infinity of f of x evaluates to one.  Screenshot.

Desmos will evaluate convergent integrals with infinite limits. Type infinity into either the upper or lower bound!

Divergent integrals with infinite limits will give a result of undefined in the expression list.

f of x is defined as one over x to the a power. a has a slider set in increments of one, running from 2 to 10. As a progresses, the integral from one to infinity of f of x evaluates to one, then one-half, then one-third, then one-fourth, and so on, all the way to one-ninth. Animated.

Use variables and sliders to build understanding of integrals with infinite limits.

 

Integrals In Action

"The best way to learn is to do." – Paul Halmos

Example graph from link below. Screenshot.

Example Graph

Explore the Fundamental Theorem of Calculus.

Example graph from link below. Screenshot.

Example Graph

Explore arc length by integration.

Example graph from link below. Screenshot.

Example Graph

Explore different integration bounds using lists, variables, and sliders.

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