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Extra care is needed to determine the intervals of θ values (e.g, [α1, β1] and [α2, β2]) over which the outer and inner boundaries of the region are traced out. Area between r = ρ1(θ) and r = ρ2(θ) Z β2 1 2 area of Ω = ρ2(θ) dθ. α2 2.
Math 20B Area between two Polar Curves. Analogous to the case of rectangular coordinates, when nding the area of an angular sector bounded by two polar curves, we must subtract the area on the inside from the area on the outside.
POLAR FUNCTIONS. From the AP Calculus BC Course Description, students in Calculus BC are required to know: The analysis of planar curves, including those given in polar form. Derivatives of polar functions. Finding the area of a region, including a region bounded by polar curves. In Precalculus students should have learned to:
Let R be the region that is inside the graph of r = 2 and also inside the graph of r = 3 + 2 cos q, as indicated above. Find the area of R. A particle moving with nonzero velocity along the polar curve given by r = 3 + 2 cos q has position (x(t), y(t)) at time t, with q = 0 when t = 0.
To determine the area of a region described by a polar equation, follow these steps: Graph the region; draw lines from the origin to the curve to determine r. If necessary, determine where the curves intersect. Set up an integral for each region with distinct outer or inner boundaries.
Related topic: The area of the region between two polar curves is given by A f g 2 d 2 1. In rectangular coordinates, we said “above” – “below”. In polar coordinates, we’ll say “outside” – “inside”.
To determine the area of a region described by a polar equation, follow these steps: 1. Graph the region 2. If necessary, determine where the curves intersect 3. Set up an integral for each region with distinct outer or inner boundaries 4. Determine the bounds of integration for each region 5. Evaluate the integrals. Examples
