Calculus Visualizations

1. Introduction

The third semester of Calculus is concerned with derivatives and integrals in 3 dimensions. Being able to visualize surfaces in 3 dimensions is helpful in understanding tangent planes, normal lines, gradient vectors and local extrema. Being able to visualize regions bounded by intersecting surfaces in 3 dimensions is essential in evaluating triple integrals. Since integrals over regions in 3 dimensions are evaluated by writing the triple integral as an iterated integral, there are six different iterated integrals that could be used. The limits of integration in these iterated integrals are obtained by projecting the region onto one of the coordinate planes. Unless one has an accurate picture of the region in 3 dimensions it is nearly impossible to determine the correct limits of integration.

This Web page presents several examples of regions in 3 dimensions to help the student visualize such regions.

2. Using the Animation tool

Each section of this Web site contains static images and animations to help the user visualize objects in 3 dimensions. All of the animations are run inside an Animation tool like the one below.

• The animation is started and stopped by clicking the Loop button. The speed of the animation can be increased or decreased by clicking the + and - buttons.
• The animation can be returned to its starting frame by clicking the Reset button.
• The animation can be viewed a frame at a time in either a forward or backward direction by clicking the < and > buttons.

3. Topics from differential calculus

• The gradient vector and level curves
• Tangent plane and normal vector to a surface
• A surface in three dimensions with a saddle point
• A function with two local maxima but no local minimum
• Space curves and T, N, B vectors.

4. Topics from integral calculus

• Solids of Revolution (about y-axis)
• Solids of Revolution (about x-axis)
• Solids of Revolution (Torus)
• Volume by cross-sections, Example 1
• Volume by cross-sections, Example 2
• The intersection of a sphere and a cylinder
• The intersection of a sphere and a cone
• Intersecting cylinders
• Intersecting cylinders and planes
• Regions defined by intersecting planes