Statics: Lesson 42 - Centroids by Calculus: Step-by-Step Triangle Example

Added: 2026-05-06

[00:00:00]What is the differential area?

[00:00:02]It's a rectangle. What is the area of a rectangle? Width times height. Okay, so the height is that.

[00:00:08]h over b times x and the width is dx, okay? But we need the integral of that. So let's integrate that and integrate that. And when you integrate something, what do you have to have? You have to have limits. From where to where, okay? So we're going to integrate from zero all the way over here, right?

[00:00:26]It's like adding up like think of these strips as like a stack of books, right?

[00:00:29]We're adding up that stack of books and it goes from zero to b, okay? From zero to b.

[00:00:35]So, can you integrate that? That's as bad as this is going to get. So yes, I can do that. I've got a constant, so I'll pull that out. h over b and what is the integral of x with respect to x?

[00:00:46]That is x squared over two from zero to b. So plug a b in for x and I get I get h b squared over two b.

[00:00:58]And then one of my b's goes away and look what I have.

[00:01:03]What have we just done? What have we just done? If you add up all the strips from here to there, you get the area. We have just proven that the area of a triangle is 1/2 base times height. I always wondered where that came from.

[00:01:14]There you go, okay? So what do we have?

[00:01:16]We have the bottom part of this equation here, right? So we know that the bottom part down here is bh over two, okay? So what is the top part? Well, the top part is the integral of x times da. So in the x direction, where is the center of each strip? Well, it's x over to that, right?

[00:01:36]So for this one, instead we already know what da is. There's da, right? That's unintegrated da right there, okay? So we're going to take unintegrated da and just multiply it by x, okay? So that's going to be the integral of x da is equal to the integral same limits zero to b of h over b.

[00:01:56]But it's x and I'm going to multiply that by x, so that just becomes x squared dx, right? Because it's x [clears throat] times da, so we just multiplied that guy times the extra x.

[00:02:06]Okay, so here we go. Let's integrate that. That becomes Here's a constant h over b and then x squared integrates to x cubed over three from zero to b, right? Plug a b in where I have an x, right? One of the b's cancels out and I get h b squared over three.

[00:02:24]Okay? So that goes up here, h b squared over three. All right, so what's going to cancel out?

[00:02:31]The h's cancel out. One of the b's goes away and the two goes up there and I'm left with x bar is equal to two thirds um b.

[00:02:42]Okay?

[00:02:43]So if you come over two thirds, there's one third, two thirds, right? So at one third the base, that's where I would put my finger in the x direction to balance that triangle on the end of my finger.

[00:02:53]That's all there is to it.

[00:02:55]Okay?