Indefinite integral and average distance

[-EquinoX-]

Homework Statement

Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph of y = x - x^2.

Homework Equations

The Attempt at a Solution

Can someone guide me how to solve this?

 

[gabbagabbahey]

Pick any point in the region...what is the distance from that point to the x-axis?

If you had a finite number of points in the region, you would add up the distance from each point to the x-axis and divide by the number of points, correct?

Well, there are of course an infinite number of points in the region, so instead of adding, you integrate...

 

[HallsofIvy]

The "average value" of any function, f(x,y) on a region R is
$$\frac{\int_R f(x,y)dxdy}{area of R}$$

 

[-EquinoX-]

what is the area of R here? I already calculate the integral part

 

[gabbagabbahey]

what is the area of R here? I already calculate the integral part

How do you usually calculate the area of a region?

 

[Mark44]

How do you normally calculate the area of a region bounded by a curve and the x-axis? Have you made a sketch of the region?

 

[-EquinoX-]

yes I have made a sketch, it's an upside down parabola right? so I should do integration to find the area?

 

[HallsofIvy]

Yes, of course. You want
[tex]\frac{\int\int_R f(x,y)dydx}{\int\int_R dydx}[/itex]
where f(x,y) is the "distance to the x-axis".

 

[-EquinoX-]

okay got it. Thanks