D dx ds 2 2 cos sin sin cos dx dr r d d dy dr r d d

Info icon This preview shows pages 29–37. Sign up to view the full content.

View Full Document Right Arrow Icon
d dx ds 2 2 + = cos sin sin cos dx dr r d d dy dr r d d θ θ θ θ θ θ θ θ = - = + From before:
Image of page 29

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Since cos 2 θ + sin 2 θ = 1. 2 2 2 2 2 2 2 2 2 2 2 2 cos 2 cos sin sin sin 2 sin cos cos dx dy d d dr dr r r d d dr dr r r d d dr r d θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ + = - + + + + = +
Image of page 30
ARC LENGTH IN POLAR COORDINATES Therefore, the length of a curve with polar equation r = f ( θ ), a θ b, is: 2 2 b a dr L r d d θ θ = +
Image of page 31

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Find the length of the polar curve r = 2cosθ Example Previously we graphed the curve, so we know the interval of integration is [0, π]
Image of page 32
2 2 b a dr L r d d θ θ = + θ θ θ π d - + = 0 2 2 ) sin 2 ( ) cos 2 ( = + = π π θ θ θ θ 0 0 2 2 2 sin 4 cos 4 d d π θ π 2 2 0 = =
Image of page 33

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Area in Polar Coordinates Sector: a slice of a circle determine by a central angle. radians in , 2 2 1 θ θ r A = Area of a sector:
Image of page 34
Using Riemann sums and the above formula when working with a polar region:
Image of page 35

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
We see that each polar rectangle has area [ ] θ θ = 2 2 1 ) ( i i f A So the area of the entire region is θ θ β α d f A A n i i n 2 2 1 1 )] ( [ lim = = =
Image of page 36
Example Find the area enclosed by the polar curve r = 2cosθ From previous work, we know that the interval needed to trace the entire curve is [0, π] θ θ π d A 2 0 2 1 ] cos 2 [ = θ θ π d = 0 2 cos 2 θ θ π d + = 0 2 ) 2 cos( 1 2 π θ θ π = + = 0 2 1 ) 2 sin(
Image of page 37
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern