Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews June 11, 2023June 4, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Polar coordinates provide an alternate way of specifying a point in the plane. The polar
coordinates [r, θ] represent the point at a distance r from the origin, rotated
θ radians counterclockwise from the positive x-axis. Since r represents a
distance, it is typically positive. Sometimes, r is allowed to be negative; in this case
[r, θ] represents the reflection about the origin of the point [| r|, θ].

It follows from basic trigonometry that the point with polar coordinates [r, θ] has
Cartesian coordinates

(r cosθ, r sinθ)

Going the other direction, the point with Cartesian coordinates (x, y) has polar
coordinates

, tan^{-1}

if it lies in quadrants I or IV and polar coordinates

, tan^{-1} + Π

if it lies in quadrants II or III.

A polar function r(θ) has a graph consisting of the points [r(θ), θ].
Such a graph is known as a polar curve. One of the simplest polar curves is the circle, the
graph of the polar function r(θ) = c, for some constant c. In the remainder of this
section, we investigate how to find the area enclosed by a polar curve from one value of
θ to another. For example, we might wish to find the area of the region below the
curve r(θ) = 1 from θ = 0 to θ = Π/2 (this region is of course a quarter
of the interior of a unit circle).

Considering the general case, the idea is similar to the idea for finding the area below the
graph of a function in Cartesian coordinates. In that case, we approximated the region by
a bunch of thin rectangles; here, we approximate it by thin circular sectors (shaped like
slices of pie).

Such a method worked before because we knew beforehand how to compute the area of a
rectangle. Now we attempt this computation for a circular sector. Suppose the sector has
angular width of Δθ and is part of a circle of radius r, with area Πr^{2}.
Since the sector accounts for Δθ/2Π of the area of the circle, the area of the
sector is equal to

Πr^{2} = (Δθ)r^{2}

Summing together the areas of all the thin sectors and taking the limit as Δθ→ 0 (and the number of sectors approaches infinity), we get the definite
integral

r(θ)^{2}dθ

Note that, because of the square in the expression being integrated, the integral counts all
area as positive, even when r(θ) < 0.

Applying this theory to the example given above, we get an area of

(1)^{2}dθ = θ =

which is indeed one quarter of the area of a unit circle.