What geometric shapes have formulas for calculating area?

How do relative (or local) minimum points and relative (or local) maximum points help someone solve an optimization problem? 

What geometric shapes have formulas for calculating area?  What kinds of shapes do not have a geometric formula for calculating area, for which someone would need calculus?

Why are rectangles used in Riemann Sums?  What are some ways that someone can get a pretty good estimate for a given area using Riemann Sums?

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