Double integral with cos(x^n) term

In summary, the conversation is about solving a double integral with a cos(x^n) term, where the integral is given and the homework statement and equations are provided. The person attempting the solution mentions trying to switch the order of integration and substituting for trigonometric identities, but is struggling. Another person points out the mistake in changing the limits and suggests using a cubic root instead of a square root.
  • #1
BilalX
7
0
[SOLVED] Double integral with cos(x^n) term

Homework Statement



Solve the following integral (without using a series development):
[tex]\displaystyle
\int _{0}^{\frac{1}{8}}\int _{\sqrt[3]{y}}^{\frac{1}{2}}\cos\left(20{\pi}x}} ^{4}\right)dx dy[/tex]

Homework Equations



N/A

The Attempt at a Solution



Obviously the cos(x^4) part is what throws me off, I tried switching the order of integration and substituting for various trigonometric identities but it doesn't seem to help much. I'd appreciate any pointers to get me started.
 
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  • #2
I think changing the order of integration is the way to go. You'll get an x^3 term in the integral with respect to x. Then it's easy.
 
  • #3
your limits should change too!
I think you did not change the limits
 
  • #4
Great, thanks, I went back and realized I'd made a stupid mistake and somehow used a square root instead of the cubic root when changing the limits.
 

1. What is a double integral with cos(x^n) term?

A double integral with cos(x^n) term is a mathematical expression that involves finding the area under a curved surface defined by the function cos(x^n). It is a type of integration that involves evaluating a double integral, which combines two single integrals into one expression.

2. How do you solve a double integral with cos(x^n) term?

To solve a double integral with cos(x^n) term, you must first convert it into a single integral by using the properties of integration. Then, you can use various techniques such as substitution, integration by parts, or trigonometric identities to evaluate the integral. Finally, you can solve the resulting equation and find the numerical value of the double integral.

3. What are the applications of double integrals with cos(x^n) term?

Double integrals with cos(x^n) term have various applications in mathematics, physics, and engineering. They are used to calculate the volume under a curved surface, to solve optimization problems, and to find the center of mass of a two-dimensional object. They are also used in probability and statistics to calculate the joint probability of two random variables.

4. What are the limitations of using double integrals with cos(x^n) term?

The main limitation of using double integrals with cos(x^n) term is that they can be challenging to evaluate analytically. In some cases, it may not be possible to find a closed form solution, and numerical methods must be used instead. Additionally, the integration limits and the function itself may be complex, making it difficult to find a solution.

5. Are there any alternative methods to solve double integrals with cos(x^n) term?

Yes, there are alternative methods to solve double integrals with cos(x^n) term. One method is to use software or calculators that have built-in integration capabilities. Another method is to use numerical integration techniques such as Simpson's rule or the trapezoidal rule. These methods provide an approximate solution to the double integral without the need for complex mathematical calculations.

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