Polar Butterfly

Area

100

Area of one of the two circles at the end of the antennae

What is 0.01π ?

200

Area of body (hint: area of an ellipse = π ab) 

Accurate to 3 decimal places

What is 0.316π ?

300

Equation for the integral of a polar

What is 1/2 times the integral from a to b of r² with respect to theta?

400

Sum of the areas of the wings

What is 240 units²?

Perimeter

100

Arc length formula

What is the integral from x₁ to x₂ of the square root of f'(x)²+1 with respect to x?

200

Length of one antenna (x₁=0.291, x₂=1.490)

What is 6.539

300

Equation for the arc length of a polar

What is the integral from angle a to angle b of the square root of (r² + (r'(θ)²))?

400

Perimeter of all four wings

What is 137.210

Basics of Polars

100

Name of polar equation that means "snail" in French

What is a limaçon?

200

Rectangular x-coordinate of a polar

What is r cos(θ)?

300

Derivative of a polar

What is (dy/dθ) / (dx/dθ)?

400

Equation for a rose curve on the x-axis

What is r=a cos(bθ)?