CODEWITHSUNDEEP
mathnumerical-integrationmaple
user3466563
user3466563

Reputation: 69

Calculating exact volume of an irregular object

For an assignment I have to use numerical integration technique to calculate volume with cylindrical surface

Ω={(x,y,z) in R³ with (x−0.5)² +(y−0.5)² ≤ 0.5² and 0 ≤ z ≤ |ln(x+y)|}.

I have used Monte Carlo technique to calculate the volume. But to be sure the answer is correct I want to check the exact volume using Maple. I have been searching online on how to do it but couldn't find it.

So the question is, is there a way to calculate exact volume using Maple for that object or integral like this:

enter image description here

Upvotes: 0

Views: 387

Answers (1)

Nico Schlömer
Nico Schlömer

Reputation: 58871

The volume is the integral of

V = int_0^1 
        int_{0.5-sqrt(0.5^2-(x-0.5)^2)}^{0.5+sqrt(0.5^2-(x-0.5)^2)} 
            int_0^{abs(log(x+y))} 
                1 dz dy dx

or, after a change of variables,

V = int_-1^1
        int_{-sqrt(1-x^2)}^{+sqrt(1-x^2)}
            0.25 * abs(log(x/2+y/2+1)) dy dx

Computer algebra systems can compute the value to 

V = 0.25502

(e.g., https://www.wolframalpha.com/input/?i=int_-1%5E1+int_%7B-sqrt(1-x%5E2)%7D%5E%7B%2Bsqrt(1-x%5E2)%7D+1%2F4+*+abs(log(x%2F2%2By%2F2%2B1))+dy+dx).

Upvotes: 0

Related Questions