❤️ area of an actual heart ❤️ @drpeyam

Uploaded February 2022 | Updated September 2026, 1 week ago

Calculating the area of an actual heart. In honor of valentine's day, I evaluate the area of a heart by using integrals under parametric curves. This much fancier than the cardioid. It involves the chain rule and some trigonometric integrals. It also explores symmetry and even and odd features of the graph, and at the end there is a nice simplification into linear terms. No polar coordinates or double integrals involved, just some nice multivariable calculus.

Area of cardioid: youtube.com/watch?v=YYpZHnrtR9g

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$Half integral of x Defining the half integral, which is halfway between a function and its integral. Done by analogy with half derivatives, by finding formulas for the derivatives of power functions. It uses the factorial which is then generalized to the gamma function with Gaussian integrals. I also mention trig functions and applications to broken processes. This is a delight for any calculus student or anyone who is interested in fractional derivatives. 0:00 Introduction 0:30 Analogy 2:58 Gamma Function 5:00 Half integral of x 9:37 Properties and Applications Half Derivative: https://youtu.be/gaAhCTDc6oA YT channel: https://youtube.com/drpeyam TikTok channel: https://www.tiktok.com/@drpeyam Instagram: https://www.instagram.com/peyamstagram/ Twitter: https://twitter.com/drpeyam Teespring merch: https://teespring.com/stores/dr-peyam$

the outstanding Laplace method for solving systems of ode