Horizontal direction: r=a( 1-cosθ) or r = a (1+cos θ) (a > 0)

Vertical direction: r=a( 1-sinθ) or r = a (1+sinθ) (a > 0)

When calculating the area of the heart-shaped line by definite integral, the graphs of 0 to π and π to 2π in the horizontal direction are symmetrical about X, so only half of the area needs to be multiplied by 2.

Extended data

The story of the heart line:

Christina was a princess in Sweden in the17th century. She is beautiful, kind and intelligent, and especially likes math. One day, she put on casual clothes and went outside the palace. On the way, she saw many beggars, one of whom was very special. He didn't take the initiative to ask passers-by for charity, but quietly squatted on the ground and concentrated on studying math problems.

The man didn't know that the lady standing in front of him was the princess, but he was surprised by the mathematical talent revealed in the lady's words, so he was happy to talk to Christina. Princess Christina realized that he was originally a mathematician, but unfortunately, for some reason, he was frustrated in doing mathematics in France and was down and out, and finally he wandered to Sweden.

So Princess Christina invited mathematicians to the palace to be her math teacher, and they discussed math problems together and talked about everything. Over time, two people are immersed in their own world of mathematics and love, and they are very happy.