Solving a Complex Integral: Finding the 4

In summary, a complex integral involves calculating the area under a complex function on the complex plane and is important in solving complicated problems in various fields. The general steps for solving a complex integral are identifying the function and limits, simplifying the function, using techniques such as substitution and integration by parts, and checking for convergence and accuracy. Common techniques for solving complex integrals include substitution, integration by parts, partial fractions, and the residue theorem. To check a solution for a complex integral, one can graph the function, use numerical methods, or apply the fundamental theorem of calculus.
  • #1
yoleven
78
1

Homework Statement


[tex]\int[/tex][tex]\int[/tex]D[tex]\left|x\right|[/tex]dA
D= X2+y2<=a2 where a>0


Homework Equations


[tex]\int[/tex][tex]\stackrel{\Pi/2}{0}[/tex][tex]\int[/tex][tex]\stackrel{a}{0}[/tex] r cos [tex]\Theta[/tex] r dr d[tex]\Theta[/tex]

I hope that's clear...

I evaluate this to [tex]\frac{a^3}{3}[/tex] sin [tex]\Theta[/tex]


sin [tex]\Pi[/tex]/2 = 1

so I get [tex]\frac{a^3}{3}[/tex]

the book says the answer is 4 [tex]\frac{a^3}{3}[/tex]



I can't see where the 4 is coming from.

can some one explain this to me?
 
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  • #2
Your domain of [0,pi/2] only includes a quarter of the entire circle.
 
  • #3
Of course. Thanks.
 

Related to Solving a Complex Integral: Finding the 4

What is a complex integral?

A complex integral is a mathematical concept that involves calculating the area under a complex function on the complex plane. It is similar to calculating the area under a curve in basic calculus, but involves complex numbers and complex functions.

Why is solving a complex integral important?

Solving a complex integral is important because it allows us to solve complicated problems in fields such as physics, engineering, and economics. It also helps us understand the behavior of complex systems and make predictions based on mathematical models.

What are the steps to solving a complex integral?

The general steps to solving a complex integral are:

  1. Identify the function and limits of integration
  2. Simplify the function if possible
  3. Use techniques such as substitution, integration by parts, or partial fractions to solve the integral
  4. Check for convergence and accuracy of the solution

What are some common techniques for solving complex integrals?

Some common techniques for solving complex integrals include:

  1. Substitution: replacing a variable with a new one to simplify the integral
  2. Integration by parts: breaking down the integral into smaller parts and applying the product rule of differentiation
  3. Partial fractions: decomposing a rational function into simpler fractions
  4. Residue theorem: using complex analysis to calculate integrals around closed paths in the complex plane

How can I check my solution for a complex integral?

To check your solution for a complex integral, you can use methods such as:

  1. Graphing the function and comparing it to the calculated value
  2. Using numerical methods to approximate the integral and comparing it to the calculated value
  3. Applying the fundamental theorem of calculus to verify the solution

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