Lengths in Polar Coordinates

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1 Lengths in Polar Coordinates Given a polar curve r = f (θ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using the parameter θ x(θ) = f (θ) cos θ y(θ) = f (θ) sin θ To compute the arc length of such a curve between θ = a and θ = b, we need to compute the integral Z b r L = ( dx dθ ) + ( dy dθ ) dθ a We can simplify this formula because ( dx dθ ) + ( dy dθ ) = (f (θ) cos θ f (θ) sin θ) + (f (θ) sin θ + f (θ) cos θ) = [f (θ)] + [f (θ)] = r + ( dr dθ ) Thus L = Z b a r r + ( dr dθ ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

2 Example 1 Compute the length of the polar curve r = sin θ for θ π Lecture 7: Areas and Lengths in Polar Coordinates

3 Example 1 Compute the length of the polar curve r = sin θ for θ π Last day, we saw that the graph of this equation is a circle of radius and as θ increases from to π, the curve traces out the circle exactly once Lecture 7: Areas and Lengths in Polar Coordinates

4 Example 1 Compute the length of the polar curve r = sin θ for θ π Last day, we saw that the graph of this equation is a circle of radius and as θ increases from to π, the curve traces out the circle exactly once We have L = R q π r + ( dr dθ ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

5 Example 1 Compute the length of the polar curve r = sin θ for θ π Last day, we saw that the graph of this equation is a circle of radius and as θ increases from to π, the curve traces out the circle exactly once We have L = R q π r + ( dr dθ ) dθ r = sin θ and dr = cos θ. dθ Lecture 7: Areas and Lengths in Polar Coordinates

6 Example 1 Compute the length of the polar curve r = sin θ for θ π Last day, we saw that the graph of this equation is a circle of radius and as θ increases from to π, the curve traces out the circle exactly once We have L = R q π r + ( dr dθ ) dθ r = sin θ and dr L = R π = cos θ. dθ p sin θ + cos θdθ = R π p sin θ + cos θdθ = π. Lecture 7: Areas and Lengths in Polar Coordinates

7 Areas in Polar Coordinates Suppose we are given a polar curve r = f (θ) and wish to calculate the area swept out by this polar curve between two given angles θ = a and θ = b. This is the region R in the picture on the left below: Dividing this shape into smaller pieces (on right) and estimating the areas of the small pieces with pie-like shapes (center picture), we get a Riemann sum approximating A = the area of R of the form nx f (θi ) A θ Letting n yields the following integral expression of the area Z b a f (θ) i=1 dθ = Z b a r dθ Lecture 7: Areas and Lengths in Polar Coordinates

8 Example Compute the area bounded by the curve r = sin θ for θ π. Lecture 7: Areas and Lengths in Polar Coordinates

9 Example Compute the area bounded by the curve r = sin θ for θ π. Using the formula A = R b a f (θ) dθ = R π (sin θ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

10 Example Compute the area bounded by the curve r = sin θ for θ π. Using the formula A = R b f (θ) dθ = R π (sin θ) a dθ R Using the half-angle formula, we get A = 1 π 1 cos(θ)dθ = ˆθ 1 sin(θ) π/ = π. 8 Lecture 7: Areas and Lengths in Polar Coordinates

11 Areas of Region between two curves If instead we consider a region bounded between two polar curves r = f (θ) and r = g(θ) then the equations becomes 1 Z b a f (θ) g(θ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

12 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ Lecture 7: Areas and Lengths in Polar Coordinates

13 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ We use the formula A = 1 R b a f (θ) g(θ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

14 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ R We use the formula A = 1 b f a (θ) g(θ) dθ To find the angles where the curves intersect, we find the values of θ where sin θ = 1 + sin θ or sin θ = 1. That is sin θ = 1/ Lecture 7: Areas and Lengths in Polar Coordinates

15 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ R We use the formula A = 1 b f a (θ) g(θ) dθ To find the angles where the curves intersect, we find the values of θ where sin θ = 1 + sin θ or sin θ = 1. That is sin θ = 1/ Therefore θ = π, 5π. Lecture 7: Areas and Lengths in Polar Coordinates

16 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ R We use the formula A = 1 b f a (θ) g(θ) dθ To find the angles where the curves intersect, we find the values of θ where sin θ = 1 + sin θ or sin θ = 1. That is sin θ = 1/ Therefore θ = π, 5π 1. R b a f (θ) g(θ) dθ = 1 R 5π π 9 sin θ (1 + sin θ) dθ Lecture 7: Areas and Lengths in Polar Coordinates

17 Example Example Find the area of the region that lies inside the circle r = sin θ and outside the cardioid r = 1 + sin θ R We use the formula A = 1 b f a (θ) g(θ) dθ To find the angles where the curves intersect, we find the values of θ where sin θ = 1 + sin θ or sin θ = 1. That is sin θ = 1/ Therefore θ = π, 5π 1. R b a f (θ) g(θ) dθ = 1 R 5π π 9 sin θ (1 + sin θ) dθ = 1 R 5π π 8 sin θ 1 sin θdθ = R 5π π sin θdθ 1 R 5π π 1dθ R 5π π = R 5π π = π. 1 cos(θ)dθ π + cos θ 5π/ π/ sin θdθ = π sin(θ) 5π/ π/ π + Lecture 7: Areas and Lengths in Polar Coordinates

18 Word of Warning NOTE The fact that a single point has many representation in polar coordinates makes it very difficult to find all the points of intersections of two polar curves. It is important to draw the two curves!!! Example Find all possible intersection points of the curves r = cos θ and r = Lecture 7: Areas and Lengths in Polar Coordinates

19 Word of Warning NOTE The fact that a single point has many representation in polar coordinates makes it very difficult to find all the points of intersections of two polar curves. It is important to draw the two curves!!! Example Find all possible intersection points of the curves r = cos θ and r = Here we must solve two equations 1 = cos(θ) and 1 = cos(θ) since the curve r = 1 is the same as the curve r = 1. Lecture 7: Areas and Lengths in Polar Coordinates

20 Word of Warning NOTE The fact that a single point has many representation in polar coordinates makes it very difficult to find all the points of intersections of two polar curves. It is important to draw the two curves!!! Example Find all possible intersection points of the curves r = cos θ and r = Here we must solve two equations 1 = cos(θ) and 1 = cos(θ) since the curve r = 1 is the same as the curve r = 1. From the first equation, we get θ = ± π, ± 5π and from the second we get θ = ± π, ± π. Lecture 7: Areas and Lengths in Polar Coordinates

21 Word of Warning NOTE The fact that a single point has many representation in polar coordinates makes it very difficult to find all the points of intersections of two polar curves. It is important to draw the two curves!!! Example Find all possible intersection points of the curves r = cos θ and r = Here we must solve two equations 1 = cos(θ) and 1 = cos(θ) since the curve r = 1 is the same as the curve r = 1. From the first equation, we get θ = ± π, ± 5π and from the second we get θ = ± π, ± π. Therefore we have 8 points of intersection θ = ± π, ± π, ± π, ± 5π. Lecture 7: Areas and Lengths in Polar Coordinates

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