Why do you calculate the area UNDER a curve with integration?

[user111_23]

I'm think it's because it's impractical to find the area outside a closed figure. But I'm still not sure.

[Office_Shredder]

Because that's how it's defined? The area outside of a closed figure is going to be infinite unless you have a very contrived scenario

[n!kofeyn]

For the same reason it doesn't make sense to find the area outside of a square, circle, or anything else.

[daniel_i_l]

One of the ways that integration of a bounded function f on the segment [a,b] defined is by dividing [a,b] into an increasingly large number of segments. Then for each segment construct a rectangle whose width is the length of the segment and whose height is the value of the function at some point in the segment. The sum of the area of these rectangles approaches the integral as the segments get smaller.
It's easy to see that by this definition the integral is equal to the area between the function and the x-axis, or, the area "under" the line.