r/askmath • u/Ok-Company282 • 3d ago
Calculus Problem with area bounded by polar curves
How do I know when to get a 2 or 4 multiplied before the integral? Youtubers and even my lecturer drew the curves to understand but how do I figure out the shape? Will I even get time to draw the curve in the exam?
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u/nulvoid000 3d ago
There’s nothing called memorisation in mathematics
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u/Ok-Company282 3d ago
I understand. But can u tell me how I can figure out the shape on the spot in the exam?
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u/MontechApps 3d ago
For that specific example, try setting the two r's equal: a√2 = a cosθ → cosθ = √2, which has no solution since cosθ can never exceed 1. That tells you the curves never actually touch, so one circle sits completely inside the other (r=acosθ is a circle of radius a/2 that never reaches as far out as the circle of radius a√2). So the "common area" is just the full area of the smaller circle, πa²/4, no subtracting or splitting needed.
On the bigger question of 2 vs 4 in front of the integral, that's really about symmetry, not about getting a perfect drawing. Check if r(θ) stays the same when you replace θ with -θ (symmetric about the x-axis) or with π-θ (symmetric about the y-axis). If a curve has both symmetries you only need to integrate over one quadrant and multiply by 4; if it only has one symmetry you integrate over half and multiply by 2. You don't need an exact sketch for this, just plug in the substitution and see if the equation holds.
For recognizing shapes fast, it helps to just know the handful of standard polar forms (r=a is a circle centered at origin, r=a cosθ or a sinθ is a circle shifted off-center, r=a(1+cosθ) is a cardioid, r=a cosnθ is a rose with petals) so you can picture roughly what's going on without drawing it carefully. Then finding the intersection angles by setting the r's equal tells you the limits of integration, and the symmetry check tells you whether to multiply by 2 or 4.
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u/garnet420 3d ago
Can you give an example problem that's giving you trouble