Showing J-Measurability of Set K with Area = Integral

[missavvy]

Homework Statement

Let f:[a,b] -> [tex]\Re[/tex] be an integrable function

Suppose f[tex]\geq[/tex]0 and let K = { (x,y) : x[tex]\in[/tex] [a,b] and 0[tex]\leq[/tex]y[tex]\leq[/tex]f(x)}

Show that K is Jordan Measurable and its area = Integral from a,b [ f(x)dx]

Homework Equations

The Attempt at a Solution

Ok so I first proved that f has zero content..
Then the set K has a boundary which is f, the a line on the x-axis on the left, and b on the right, is that correct? and of course the x-axis itself since y[tex]\geq[/tex]0.

Then if I take the Union of f, a, b (by these points i mean x=a,x=b) and the x-axis.. i can prove they have zero content so K is measurable.

Is that first part ok?

 

[lippyka]

For the second part I think we just have to use Fubini's theorem and calculate the area by double integral.. but how? how do i set it up?