Calculating areas bounded by polar curves looks extremely difficult. Do Americans really need to integrate such a complex expressions without a calculator?

Question

I am a Japanese and not familiar with a calculus course in the United States.
Looking around Socratic, I found there are so many questions about areas bounded by a polar curve, but few answers are posted.

I know the formula #S=1/2int_alpha^beta{f(theta)}^2d##theta#.
However, this integration looks often beyond our ability, such as in my previous post which I have finally given up.

Some questions seem even more difficult. I wonder how the students in the USA perform such a complex integration without a calculator.

Charles Anctil · Accepted Answer

I'm really not sure who's asking these questions. The linked question is a great example of one.

As an American high schooler, I've never been asked such a complex question for school. I think the goal with those is just to use a calculator. Maybe they're just testing knowledge of the polar area formula. Maybe they're testing calculator skills. Not really sure. I'm still gonna try my best to do the linked problem, though!

Ezra Adams · Answer

undefined Yes, Americans (and anyone else studying mathematics) do need to integrate complex expressions involving polar curves without a calculator, particularly in educational settings such as high school or university-level calculus courses. Understanding how to calculate areas bounded by polar curves is an important part of learning calculus and mathematical problem-solving skills. While it may seem difficult at first, with practice and understanding of the underlying principles, students can develop the ability to solve such problems manually. Integrating complex expressions without a calculator is a fundamental aspect of mathematical education that helps students develop critical thinking, problem-solving, and mathematical reasoning abilities.