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