There is a series of tricks to prove some inequalities, using geometric method about area comparison.
: We interpret 
and 
as areas of some regions. These areas should be easy to compare.
A toy example: consider a square inscribe on a unit circle. The area of the unit disc is 
, and the area of this square is 
. Consequently, 
.
Another example: Theorem 6.1. By observation, 
%20%5C%2C%20dx%20-%20%5Cint_a%5Eb%20g(x)%20%5C%2C%20dx#card=math&code=A%20%3D%20%5Cint_a%5Eb%20f%20%28x%29%20%5C%2C%20dx%20-%20%5Cint_a%5Eb%20g%28x%29%20%5C%2C%20dx).
Yet another example: 
%5E4%7D%7B1%2Bx%5E2%7D%20%3D%20%5Cfrac%7B22%7D%7B7%7D%20-%20%5Cpi%20%3E0#card=math&code=%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cfrac%7Bx%5E4%281-x%29%5E4%7D%7B1%2Bx%5E2%7D%20%3D%20%5Cfrac%7B22%7D%7B7%7D%20-%20%5Cpi%20%3E0).
%3D0%2C%5Cquad%20%20f(x)%20%3D%20%5Cfrac%7Bx%5E4(1-x)%5E4%7D%7B1%2Bx%5E2%7D%20%5Cgeq%200#card=math&code=g%28x%29%3D0%2C%5Cquad%20%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E4%281-x%29%5E4%7D%7B1%2Bx%5E2%7D%20%5Cgeq%200). and 
%3D0#card=math&code=f%28x%29%3D0) if and only if 
.
%20%5C%2C%20dx%20%3E0#card=math&code=%5Cint_0%5E1%20f%28x%29%20%5C%2C%20dx%20%3E0). Conclusion: 
.
X-type region: bounded left and right by two vertical lines: 
, bounded up and below by two graphs 
%2C%20y%3Dg(x)#card=math&code=y%3Df%28x%29%2C%20y%3Dg%28x%29).
Y-type region: bounded up and below by two horizontal lines: 
, bounded left and right by two graphs: 
%2C%20x%20%3D%20v(y)#card=math&code=x%20%3D%20u%28y%29%2C%20x%20%3D%20v%28y%29).
Arc length
When considering definite integrals, we require that 
must be continuous over 
.
Now we require that is differentiable over , and moreover, its derivative 
is continuous on ——> such a function is called smooth.
the length of a small arc over a sub-interval (sub-interval is obtained by partion of the total interval ) can be approximately expressed as the length of the line segment joining 
)#card=math&code=%28x%7Bi-1%7D%2C%20f%28x%7Bi-1%7D%29%29) and 
)#card=math&code=%28xi%2C%20f%28x_i%29%29). The length of this line segment can be computed via the Pythagorean theorm. This length is approximately 
)%5E2%7D%20%5CDelta%20x#card=math&code=%5Csqrt%7B1%2B%28f%27%28x_i%5E%2A%29%29%5E2%7D%20%5CDelta%20x) for some 
lie between #card=math&code=%28x%7Bi-1%7D%2C%20xi%29), by virtue of the mean value theorem. Now we sum them up, and get a Riemann-sum: )%5E2%7D%20%5CDelta%20x#card=math&code=%5Csum%7Bi%3D1%7D%5En%20%5Csqrt%7B1%2B%28f%27%28x_i%5E%2A%29%29%5E2%7D%20%5CDelta%20x).
Remark 1. A function being smooth in this textbook means that is differentiable over and is continuous over . In other popular literatures, this may be called that is a 
function, or 
. 
means is continuous.
Remark 2. Arc length, area, as well as volume, these two are geometric information (or amount/value), these amounts will not change even if we move or rotate the coordinating system.
Surface area of a revolution object 
%5Csqrt%7B1%2B(f’(x))%5E2%7D)%20dx#card=math&code=%5Cint_a%5Eb%20%282%5Cpi%20f%28x%29%5Csqrt%7B1%2B%28f%27%28x%29%29%5E2%7D%29%20dx)
Example 6.21 
, %3D%5Csqrt%7Bx%7D#card=math&code=f%28x%29%3D%5Csqrt%7Bx%7D). 
%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D#card=math&code=f%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D).
Another example: Gabriel’s horn: 
%3D1%2Fx#card=math&code=a%3D1%2C%20b%20%3D%5Cinfty%2C%20f%28x%29%3D1%2Fx). %3D-1%2Fx%5E2#card=math&code=f%27%28x%29%3D-1%2Fx%5E2).
Its surface is 
%5Cfrac%7B1%7D%7Bx%7D%20%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D%20dx#card=math&code=%5Cint_1%5E%7B%5Cinfty%7D%20%282%5Cpi%29%5Cfrac%7B1%7D%7Bx%7D%20%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D%20dx).
It can be shown that an anti-derivative is 
%20-%20%5Cln%20(%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D-1)%20%5Cbigg)#card=math&code=%5Cfrac%7B1%7D%7B4%7D%5Cbigg%28-2%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D%2B%20%5Cln%20%28%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D%2B1%29%20-%20%5Cln%20%28%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D-1%29%20%5Cbigg%29). Since 
%20%3D%20%2B%5Cinfty#card=math&code=%5Clim_%7Bx%5Cto%20%5Cinfty%7D%20-%5Cln%20%28%5Csqrt%7B1%2B%5Cfrac%7B1%7D%7Bx%5E4%7D%7D-1%29%20%3D%20%2B%5Cinfty), we deduce that the surface area is infinite.
