Motivation and Polar Coordinates in the Calculus ll | MATH 211, Assignments of Calculus

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Calculus and Polar Coordinates

MATH 211, Calculus II

J. Robert Buchanan

Department of Mathematics

Summer 2008

Motivation

Today we will focus on extending the ideas of slope, equation of the tangent line, arc length, and area to curves that are described as equations in polar coordinates.

Example

Example Find a formula for the slope of the tangent line to the graph of r = 3 − 4 sin θ.

-4 -2 2 4

Example

Example Find the points at which the graph of r = 5 − 5 sin θ has horizontal tangent lines.

-6 -4 -2 2 4 6

Area of a Circular Sector

Partition the interval [α, β] into n equal subintervals where ∆θ = β − α n

and θ k = α + k ∆θ for k = 0 , 1 ,... , n.

The area of the region in the subinterval [θ k − 1 , θ k ] can be approximated by the area of a circular sector.

Θk Θk- 1

r=fHΘL

Riemann Sum

Area of a circular sector: ∆ Ak =

r^2 ∆θ ≈

[ f (θ k )]^2 ∆θ.

A ≈

∑^ n

k = 1

[ f (θ k )]^2 ∆θ

= (^) n lim→∞

∑^ n

k = 1

[ f (θ k )]^2 ∆θ

∫ (^) β

α

[ f (θ)]^2 d θ

Example

Example Find the area enclosed by the rose: r = sin 2θ.

-0.75 -0.5 -0.25 0.25 0.5 0.

-0.

-0.

-0.

Example

Example Find the area inside the circle r = sin θ and outside the cardioid r = 1 + cos θ.

-0.5 0.5 1 1.5 2

-0.

1

Example

Example Find the perimeter of the cardioid r = a ( 1 + cos θ) where a > 0.

a

• a

a

Example

Example Find the arc length of the exponential spiral r = e θ/^2 for π/ 2 ≤ θ ≤ π.

-4 -3 -2 -

1.25^ 1.

2.25 2