Chap10_Sec4

# Chap10_Sec4 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

This preview shows pages 1–13. Sign up to view the full content.

PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES AND POLAR COORDINATES 10

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

View Full Document
10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area of a region whose boundary is given by a polar equation. PARAMETRIC EQUATIONS & POLAR COORDINATES
AREAS IN POLAR COORDINATES We need to use the formula for the area of a sector of a circle A = ½ r 2 θ where: r is the radius. θ is the radian measure of the central angle. Formula 1

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

View Full Document
AREAS IN POLAR COORDINATES Formula 1 follows from the fact that the area of a sector is proportional to its central angle: A = ( θ/ 2 π ) πr 2 = ½ r 2 θ
AREAS IN POLAR COORDINATES Let R be the region bounded by the polar curve r = f ( θ ) and by the rays θ = a and θ = b, where: f is a positive continuous function. 0 < b – a ≤ 2 π

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

View Full Document
AREAS IN POLAR COORDINATES We divide the interval [ a, b ] into subintervals with endpoints θ 0 , θ 1 , θ 2 , …, θ n , and equal width ∆θ . Then, the rays θ = θ i divide R   into smaller regions with central angle ∆θ = θ i θ i –1 .
AREAS IN POLAR COORDINATES If we choose θ i * in the i th subinterval [ θ i –1 , θ i ] then the area ∆A i of the i th region is the area of the sector of a circle with central angle ∆θ and radius f ( θ* ).

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

View Full Document
AREAS IN POLAR COORDINATES Thus, from Formula 1, we have: A i ½[ f ( θ i *)] 2 θ So, an approximation to the total area A of R   is: * 2 1 2 1 [ ( )] n i i A f θ = Formula 2
AREAS IN POLAR COORDINATES It appears that the approximation in Formula 2 improves as n .

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

View Full Document
AREAS IN POLAR COORDINATES However, the sums in Formula 2 are Riemann sums for the function g ( θ ) = ½[ f ( θ )] 2 . So, * 2 2 1 1 2 2 1 lim [ ( )] [ ( )] n b i a n i f f d θ →∞ = ∆ =
AREAS IN POLAR COORDINATES Therefore, it appears plausible—and can, in fact, be proved—that the formula for the area A of the polar region R is: 2 1 2 [ ( )] b a A f d θ = Formula 3

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

View Full Document
Formula 3 is often written as with the understanding that r = f ( θ ). Note the similarity between Formulas 1 and 4.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

### Page1 / 48

Chap10_Sec4 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online