Idea 1:

Convert polar coordinates to Cartesian coordinates:

Let any point on the image be (x, y), then:

X=r*cosθ=( 1+ cosθ)* cosθ

Y=r*sinθ=( 1+ cosθ)* sinθ

If we can get the form of y=f(x), then the problem will be transformed into a calculus problem in Cartesian coordinates-calculating the path of the curve in a certain interval, but the result is extremely cumbersome-and it is conceivable that it will eventually fail.

X = cosθ+cosθ 2, and the solution of cos θ can be obtained (a quadratic equation with one variable).

X 2+Y 2 eliminates sinθ, and then substitutes it into the solution of cosθ in the previous step to get the form of y=f(x)-it may be an implicit function, which is difficult to simplify and needs to be classified and discussed.

And then integrate from x=a to b?

Idea 2:

The combination of number and shape is intuitive and concise, but some steps are not rigorous and need to be proved by limit law. High school doesn't seem to have this requirement.

The result can be calculated by Pythagorean theorem+trigonometric function rule+basic calculus rule.

I will go to Peking University to continue my studies recently. I have encountered this problem and I am interested in trying to solve it. But I graduated from high school for a long time and studied medicine. Some terms may be incorrect, and I basically remember all the formulas used.

On the whole, it feels as difficult as the math problem in Jiangsu high school; At that time, high school mathematics calculus and polar coordinates were elective courses in Jiangsu science, and the theoretical basis needed for the topic could provide reference.