Math252SinSquaredTheta

Math252SinSquaredTheta - How do you graph the surface z =...

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How do you graph the surface z = sin 2 θ over the annulus R from our example in the 17.3 notes? How would you graph the corresponding solid whose volume we were finding? First off, let’s consider the graph of sin 2 θ vs. θ in Cartesian coordinates: The square of a real number in [0,1] will also be in [0,1]. Because the range of sin θ is [0,1], the range of sin 2 θ is also [0,1]. Unlike the graph for sin θ , there are no corners at θ -values of 0, π , 2 π , etc. The sin 2 θ function is everywhere differentiable! Its derivative is given by 2sin θ cos θ , or sin 2 θ ( ) , which is 0 at θ = π n n integer ( ) . How do you graph z = sin 2 θ in 3-space? Here’s what Mathematica gives; the coordinate axes are rotated a bit differently from what we’re used to.
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How do we get the piece over R ? You could take scissors to the previous graph, and rotate the scissors as you cut. It's basically like a twisty slide, or a piece of a roller coaster. Imagine a staircase in a mansion that is curving upward, except that we smooth out the steps. Notice that the function values are constant along the straight lines in
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