Calculus: Single Variable Part 4 – Applications

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• Computing Areas and Volumes
• Having seen some calculus before, you may recall some of the motivations for integrals arising from area computations. We will review those classical applications, while introducing the core idea of this module -- a differential element. By computing area and volume elements, we will see how to tackle tough geometry problems in a principled manner.
• Other Geometric Applications
• There's more to geometry than just area and volume! In this module, we will take things "to the next level", ascending to higher dimensions. Coming back to the 3-d world, we will return to problems of length and area, but this time in the context of curves and surfaces. As always, the emphasis will be on how to construct the appropriate differential element for integrating.
• Physical Applications
• There is so much more to applications of integrals than geometry! So many subjects, from physics to finance, have, at heart, the need for setting up and computing definite integrals. In this short but intense module, we will cover applications including work, force, torque, mass, and present & future value.
• Averages and Mass
• There is a statistical aspect to integrals that has not yet been brought up in this course: integrals are ideal for computing averages. Motivated by physical problems of mass, centroid, and moments of inertia, we will cover applications of integrals to averages.
• An Introduction to Probability
• This capstone module gives a very brief introduction to probability, using what we know about integrals and differential elements. Beginning with common-sense uniform probabilities, we move on to define probability density functions and the corresponding probability element. Building on the physical intuition obtained from centers of mass and moments of inertia, we offer a unique perspective on expectation, variance, and standard deviation.