Applications of integration: Area and boundaries? (can't understand so

[Poppietje]

Hey! I'm a complete newbie to integral calculus (and well, to math in general - but I'm trying to learn!) and I have a bit of a problem. I already get the feeling that the solution is ridiculously simple, but my brain just isn't making the connection.

Homework Statement

Given are two functions: f(x) = 2x√x and g(x) = -2x + 24 that intersect at points (4, 16)
Problem: Calculate the center of gravity bounded by the two axes and graph of g(x).

Homework Equations

The teacher's solution gives the equation A: 1/2 * 12 * 24 = 144 and the integral boundaries are defined from 0 to 12.

The Attempt at a Solution

Well, the answer is known but my problem is that I can't figure out how it got there. I do know that if an area is bounded by a graph of a function and an axis, then one of the boundaries is set at 0... But I feel that I'm missing something very basic (which wouldn't surprise me, since my math education until now is pretty, uh, bad.)
I guess the problem is that I don't know how to actually *think* about the problem, I'm just aimlessly playing around with the numbers (16 - 4 makes 12! x is 12! Is that the boundary? Why? I don't know!)

If you could kick me in the right direction I would appreciate it a lot! ^^ Thank you.

 

[BvU]

Hello Pop and welcome to PF.

You're not all that new to math: if you can count, add and subtract, that's quite a bit already!

Some general advice: if confronted with a big problem, chop it up into small pieces and start with one of them.
In this case, f(x) doesn't appear in the actual question. Only g. You can plot it easily: straight line. x=0 gives (0, 24) g=0 gives x = 12, so (12, 0).

You are asked to calculate the center of gravity. What is a releveant equation when you have to do that?

 

[Poppietje]

Hey, thank you a lot for your reply! :)

I realized, indeed, that after plotting out the graph, the area in question becomes the area of a triangle between the x- and y-coordinates. Additionally the boundaries became clear as well. And that answers my big question.

This then allows me to proceed with the problem, solving the coordinates for the center of gravity by the known equations for x_z and y_z. :) This part is no problem, just integration.

Thank you again so much! This was just what I needed.

 

[BvU]

You're welcome