Can The Following Double Integral Be Evaluated or Can It Only Be Approximated Numerically

Question

∫∫ √(4v² + 4u² + 1) dvdu, with limits of integration 0≤v≤1 & 0≤u≤2

*Note: the entire expression is supposed to be under the square root symbol

 

I have tried every integration technique I can think of to try and evaluate this this double integral, and I cannot figure it out.  I was wondering if it is even possible to explicitly evaluate the the integral (using techniques from a typical Calculus III class) or can it not be evaluated/requires higher level math techniques.  Thanks!

Answer 1

Let x = 2u and y = 2v.

 

Then (1/4) ∫∫ √(x² + y² + 1) dx dy is your transformed integral.  Looking at this you might think of polar coordinates.

 

 

Then using polar coordinates:

 

x² + y² = r² , dx dy = r dr dθ.

 

(1/4) ∫∫ √(r² + 1) r dr dθ

 

This looks a lot better.

 

Now, transform your limits using x = r cos(θ) and y = r sin(θ).  Good luck.