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Large-Scale Pastel Drawings of Endangered Icebergs by Zaria Forman

Why is the Circumference of a Circle the Derivative of its Area?: A Visual Explanation

Circumference = 2πr

Area = πr^2

You may have noticed that the circumference of circle is the derivative of its area with respect to the radius. Similarly, a sphere’s surface area (SA = 4πr^2) is the derivative of its volume (Volume = 4/3πr^3). This isn’t a coincidence! But why? And is there an intuitive way of thinking about it?

Calculus refresher: Finding the derivative of a function is finding its rate of change. For example, consider the function y = x^2. The derivative of this function is 2x, which describes how much, in terms of x, y changes when x changes. Integration is the reverse process of derivation. Finding the integral of a function first considers the function a rate of change. Then, by multiplying it by infinitesimally small increments of x from a lower bound to an upper bound, the process of integration computes the definite integral, a new function whose derivative was the original function. Think of a car moving at a velocity over time. The rate of change of the velocity is the cars acceleration. Additionally, if you multiply the velocity by how much time has passed, you get the total distance traveled by the car. Therefore, acceleration is velocity’s derivative and distance traveled is velocity’s integral.

So what is the rate of change of a circle? Consider a circle with the radius r. If you increase the radius by ∆r, the area of the new circle is πr^2 + the area of the added ring. The ring’s area is 2πr (which is the rings length) * ∆r (the ring’s height). This is indicated by the first gif, in which the new rings have the length ∆r. To find the rate of change, we take the limit as ∆r goes to 0. The limit as ∆r goes to 0 of 2πr∆r is simply 2πr! 

Let’s find the area of a circle with radius r by integrating its 2πr, its circumference. For the lower bound of our integration, think of the smallest circle we can make—a circle with radius 0. The largest circle we can make is a circle with radius r—our upper bound. We draw our smallest circle (radius 0), and then continuously add tiny rings to it by increasing r and drawing another circle, keeping the change of r as tiny as possible. We stop when r has reached our upper bound. As the second gif demonstrates, we are left with what is pretty much a filled circle! We went from 1 dimensional lines, to a 2D figure with an area of πr^2. This a fun way of visualizing the integration of 2πr from 0 to r! 

So, based on this explanation, can you figure out a way to visualize why the surface area of a sphere is the derivative of its volume? Hint: jawbreakers (or onions, alternatively)! 

Le fabuleux destin d'Amélie Poulain (2001)  dir. Jean-Pierre Jeunet 

beach dreams

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Do you think I am an automaton? — a machine without feelings? and can bear to have my morsel of bread snatched from my lips, and my drop of living water dashed from my cup? Do you think, because I am poor, obscure, plain, and little, I am soulless and heartless? You think wrong! — I have as much soul as you — and full as much heart! And if God had gifted me with some beauty and much wealth, I should have made it as hard for you to leave me, as it is now for me to leave you. I am not talking to you now through the medium of custom, conventionalities, nor even of mortal flesh: it is my spirit that addresses your spirit; just as if both had passed through the grave, and we stood at God’s feet, equal — as we are!

Charlotte Brontë, Jane Eyre (via wonderwarhol)

“I must say, I find that girl utterly delightful. Flat as a board, enormous birthmark the shape of Mexico over half her face, sweating for hours on end in that sweltering kitchen, while Mendl, genius though he is, looms over her like a hulking gorilla. Yet without question, without fail, always and invariably, she’s exceedingly lovely.”

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Most popular paintings on the blog in 2017, in no particular order:

Hilda Hechle (British, 1886 - 1939): A moonlight phantasy

Emma Ciardi (Italian, 1879 - 1933): Courtly Company with Parasols 

Elizabeth Strong (American, 1855 - 1941): Deer in the woods

Maria Wiik (Finnish, 1853 - 1928): La Polonaise (Marie Bashkirstseff) (1878)

Helmi Biese (Finnish, 1867 - 1933): View from Pyynikki Ridge (1900)

Lilian Stannard (British, 1877 - 1944): Michaelmas daisies

Isabel Codrington (British, 1874 - 1943): Evening 

Evelyn De Morgan (English, 1855 - 1919):  Aurora triumphans (1873) 

Mary Hayllar (British, active 1880 - 1885): Breakfast (1880) 

Marguerite Gérard (French, 1761 - 1837): La toilette de minette

Grand Budapest Hotel (2014) dir. Wes Anderson