How torque and integrals helped Elon Musk build Tesla

I am quite sure Musk really did use torque in building his multibillion-dollar empire. I drive a Tesla Model Y, and its door handles are cleverly hidden inside the door frame.

Without a visible handle, users have to push on the door surface to close it. Compared with the Lexus NX, another compact SUV I used for a while, closing the Tesla door feels noticeably more awkward.

Because there is no handle, I have to push with my not-so-strong arms.

A dark-blue Tesla Model S. Photo by Pexels

Calculating torque is essential in car-door design. Tesla's design prioritizes a smooth exterior body to minimize drag and improve energy efficiency, meaning the traditional handle had to be "sacrificed."

To compensate, the door's size and leverage are adjusted to increase torque and reduce the force needed to shut it.

On the other hand, Elon Musk also had to use integrals when designing Tesla cars. Air resistance while driving depends on the car's surface area and how its exterior curves are distributed.

In simple terms, steep slopes on the car body face different wind pressure than flatter areas. The friction caused by air depends on the total area of the car's surface. But how do you calculate the area of a curved, uneven surface?

To determine the total area of such a surface, you need to use integration. Integration is essentially the reverse of differentiation. When you have a non-flat surface represented by an equation, differentiation lets you break it into tiny rectangular sections. Then, by summing up all those little areas through integration, you get the total surface area of the irregular shape.

This can be illustrated as follows: The space under a graph, marked by two blue vertical lines, represents a single interval. That small area can be found using the rectangle area formula. Add up all such sections, and you get the total area under the curve. That's what integration does.

The shape of that curve resembles the contours of a car door. Using differentiation and integration helps engineers estimate the door's surface area, calculate air friction, and evaluate how drag affects energy consumption.

That's what learning calculus is for. Modern software like MATLAB can perform these calculations much faster than we can, but you still need to understand the fundamentals, like which equations to use, what to measure, and how to model a surface accurately.

Students in specialized math programs often complain that math is only useful for academic research, not for "normal people."

But I graduated as a chemical engineer, and when I had to calculate the surface area of a chemical pipe system, I still needed integration. Of course, software does most of it today, but any engineer still needs to understand how these tools work.

I can't help but laugh remembering my math teacher trying to explain why differentiation and integration matter. Everything I’ve mentioned about cubic equations and calculus wasn’t mine. It actually came from her lessons back in Vietnam, when a student asked: "What’s the point of learning calculus?"

If few engineers ever get the chance to design machines, industrial tools, or manufacturing systems, then it’s no wonder most people don’t realize what mathematics is really for.