Calculus 10.3 day 1

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10.3 Polar Coordinates
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Converting Polar to Rectangular
Use the polar-rectangular conversion formulas to
show that the polar graph of r 4 sin is a
circle.

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Converting Polar to Rectangular
Use the polar-rectangular conversion formulas to
show that the polar graph of r 4 sin is a
circle.
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One way to give someone directions is to tell
them to go three blocks East and five blocks
South.
Another way to give directions is to point and
say Go a half mile in that direction.
Polar graphing is like the second method of
giving directions. Each point is determined by a
distance and an angle.
A polar coordinate pair
Initial ray
determines the location of a point.
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Some curves are easier to describe with polar
coordinates
(Circle centered at the origin)
(Line through the origin)
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More than one coordinate pair can refer to the
same point.
All of the polar coordinates of this point are
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Tests for Symmetry
x-axis If (r, q) is on the graph,
so is (r, -q).
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Tests for Symmetry
y-axis If (r, q) is on the graph,
so is (r, p-q)
or (-r, -q).
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Tests for Symmetry
origin If (r, q) is on the graph,
so is (-r, q)
or (r, qp) .
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Tests for Symmetry
If a graph has two symmetries, then it has all
three
p
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To find the slope of a polar curve
We use the product rule here.
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To find the slope of a polar curve
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Example
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Finding slope of a polar curve
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Finding slope of a polar curve
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Area Inside a Polar Graph
The length of an arc (in a circle) is given by
r.q when q is given in radians.
For a very small q, the curve could be
approximated by a straight line and the area
could be found using the triangle formula
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We can use this to find the area inside a polar
graph.
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Example Find the area enclosed by
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Notes
To find the area between curves, subtract
Just like finding the areas between Cartesian
curves, establish limits of integration where the
curves cross.
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Finding Area Between Curves
Find the area of the region that lies inside the
circle r 1 and outside the cardioid r 1
cos Ø.
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Finding Area Between Curves
Find the area of the region that lies inside the
circle r 1 and outside the cardioid r 1
cos .
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When finding area, negative values of r cancel
out
Area of one leaf times 4
Area of four leaves
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To find the length of a curve
Remember
For polar graphs
If we find derivatives and plug them into the
formula, we (eventually) get
So
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There is also a surface area equation similar to
the others we are already familiar with
When rotated about the x-axis
p
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