Category Archives: Integral

Introduction to Double Integration

1. Consider . What is the volume of the region (1) The first thing we do: consider the rectangle and form a partition of into segments and into segments. This gives us a mesh of rectangles as specified by the … Continue reading

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Differentiation Under the Integral Sign

1. Sometimes we work with tricky integrals. For example, consider the integral (1). What is ??? We can evaluate (1) directly as (2). Thus its derivative is (3). But what about the general case?

Posted in Calculus, Differential, Integral, Integration Techniques | 2 Comments

Calculus isn’t useless: When Velociraptors Attack

1. Some critics may sneer calculus is useless. Oh yeah? Well, we can solve this problem: Question: The velociraptor spots you away and attacks, accelerating at from a stand start, up to its top speed of . When it spots … Continue reading

Posted in Applications, Calculus, Differential, Integral, Optimization | 2 Comments

Area of a Surface of Revolution (Part 1)

1. Suppose we consider some function on the domain . For example: We revolve this about the axis by 360 degrees. So this looks like Question: what is the area occupied by this surface?

Posted in Applications, Calculus, Definite Integral, Integral | 2 Comments

Calculating Arc-Length of Curves

1. Recall we have some curve usually of the form . But sometimes we can write it as where both coordinates are a function of “time” . Convention: we will set . [Caveat: some exercises may have a different range … Continue reading

Posted in Applications, Calculus, Integral | 2 Comments

Integration Technique #2: Integration By Parts

1. Try performing the following integral (1) ??? You cannot do this (easily) with the techniques we’ve discussed so far. So lets introduce a new one: integration by parts.

Posted in Calculus, Integral, Integration Techniques | 5 Comments

Integration Technique #1: Substitution

1. Try integrating (1)??? It’s kind of hard, so we try to do something very clever: use the chain rule.

Posted in Calculus, Definite Integral, Integral, Integration Techniques | 3 Comments