| 1. Known Integral: Integral of Sine, Cosine, Tangent, etc..
2. U-Substitution:
Reduction Formulas:
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Polar Curves:
| Definition. A point P in the plane has polar coordinates (r, q) if the line segment OP has length r and the angle that OP makes with the positive axis is q (measured in a counter clockwise direction).
This definition requires that r > 0. If r < 0, then we consider the point Q which has polar coordinates (-r, q). Then the point P has polar coordinates (r, q) if P is the point on the straight line containing O and Q which is -r units from O on the opposite side of O from Q.
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| Theorem. If a point P has polar coordinates (r, q) then the rectangular coordinates of P is (r cos(q), r sin(q)). In other words,
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| Theorem. If a point P has rectangular coordinates (x, y) then the polar coordinates of P is (r, q) where
r2 = x2 + y2
q = arctan(y/x). |
We choose the positive square root of x2 + y2 for r if x > 0 and the negative square root otherwise. | |
Areas Bounded By Polar Curves:
Definition. Suppose that r = f(q) is a positive continuous function defined on the closed interval [a, b] where 0 < b - a < 2p. The area bounded by the graph of r = f(q), q = a, and q = b is
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Parametric Equations in the plane is a pair of functions
x = f(t) and y = g(t)
which describe the x and y coordinates of the graph of some curve in the plane.
Differentiable Parametric Functions
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