Integral of sin3x from -a to a: Is My Answer Right?

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In summary, the formula for calculating the integral of sin3x from -a to a is ∫sin3x dx = (-1/3)cos3x + C, where C is the constant of integration. This integral can be solved using the fundamental theorem of calculus as long as the limits of integration are defined and the function is continuous. It is possible to approximate the value of this integral using numerical integration methods, and to check the correctness of the answer by taking the derivative of the calculated integral or using online integral calculators. Special cases and conditions to consider when solving this integral include defining the limits of integration, ensuring continuity of the function, and adding the constant of integration to the final answer.
  • #1
PrudensOptimus
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Take the integral of:

sin3x, with respect to x. [-a,a] - interval

I end up getting 0/3:

= [-cos(3x)/3]

= [-cos(3a) - (- cos (-3a))]/3

= [-cos (3a) + cos (3a)]/3 ---> cos(-3a) = cos 3a

= 0/3 = 0

I think I did something wrong, right?
 
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  • #2
No you are correct , sine is an odd function and integrating it over any interval symmetric about the origin will give you 0.
 
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Yes, your answer is correct. The integral of sin3x from -a to a is indeed 0. Your steps are also correct, and you did not make any mistakes. Well done!
 

FAQ: Integral of sin3x from -a to a: Is My Answer Right?

What is the formula for calculating the integral of sin3x from -a to a?

The formula for calculating the integral of sin3x from -a to a is ∫sin3x dx = (-1/3)cos3x + C, where C is the constant of integration.

Can the integral of sin3x from -a to a be solved using the fundamental theorem of calculus?

Yes, the integral of sin3x from -a to a can be solved using the fundamental theorem of calculus, as long as the limits of integration are defined and the function is continuous.

Is it possible to approximate the value of the integral of sin3x from -a to a?

Yes, it is possible to approximate the value of the integral of sin3x from -a to a using numerical integration methods such as the trapezoidal rule or Simpson's rule.

How can I check if my answer for the integral of sin3x from -a to a is correct?

You can check if your answer is correct by taking the derivative of your calculated integral and seeing if it matches the original function, sin3x. You can also use online integral calculators to verify your answer.

Are there any special cases or conditions to consider when solving the integral of sin3x from -a to a?

Yes, when solving the integral of sin3x from -a to a, it is important to consider the limits of integration and make sure they are defined, as well as the continuity of the function. It is also important to remember to add the constant of integration when finding the final answer.

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