In polar form the area is swept out by thin wedges, each ½r² dθ. Between two curves subtract the inner r² from the outer, and the hard part is the limits: where the curves meet, or where r passes through zero.
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Find the area inside r=3sinθ and outside r=1+sinθ.
The curves meet where 3 sin θ = 1 + sin θ, so sin θ = ½: θ = π/6 and θ = 5π/6.
A = ½ ∫ from π/6 to 5π/6 of (9 sin²θ - (1 + sin θ)²) dθ = ½ ∫ (8 sin²θ - 2 sin θ - 1) dθ.
Integrating and evaluating gives π.
The answer is π.
Where students go wrong
Integrating over 0 to 2π by habit. A circle like r = 3 sin θ is traced once on 0 to π; going to 2π covers it twice and doubles the area.
Also called polar area, area between polar curves, area of a region in polar coordinates or area enclosed by a polar curve.
What you can put on this worksheet
Area inside a polar curve
Find the area inside r=6sinθ→ 9π
Sectors, and between or inside two curves
Find the area inside r=5 and outside r=4.→ 9π
Lemniscates, spirals, petals and inner loops
Find the area inside both loops of r2=sin2θ.→ 1
Questions about these worksheets
Yes. Every area in polar coordinates sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the area in polar coordinates questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 3: area inside a polar curve, sectors, and between or inside two curves and lemniscates, spirals, petals and inner loops. Tick as many as you want and set how many of each, or let it spread them evenly.
Integrating over 0 to 2π by habit. A circle like r = 3 sin θ is traced once on 0 to π; going to 2π covers it twice and doubles the area.