Help with this Definite Integral

In summary, the conversation discusses the process of integrating a given function within the interval of [0,2pi], specifically in the second quadrant. The person mentions simplifying the integral and breaking it into four parts, and asks for clarification on the function in the second quadrant. The expert provides a hint that involves the positive sine and negative cosine in that quadrant.
  • #1
chuachinghong
8
0
Well as I am practising for my coming test, I encountered this question:

Integrate
f(x) = absolute [(sin X)^3 * (cos X)^15]dx
within the interval of [0,2pi]


I tried simplifying this integral into...


f(x) = absolute[ ((cos X)^15)*(sin X) -((cos X)^17)*(sin X))] dx
within the interval of [0,2pi]

Also, I break up the integral into 4 parts of pi/2 each i.e [0,pi/2], [pi/2,pi], [pi,3pi/2] and [3pi/2, 2pi]

My question is, when I am calculating the interval of the Second Quadrant, which is from [pi/2, pi]. What should my f(x) look like after taking out the absolute sign?


Hope you can help me with this. Thank you :smile:
 
Last edited:
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  • #2
sine is positive, cosine is negative in the second quadrant, and your powers are odd, so...?
 
  • #3
Oh ok I got it. Thanks for your help. @,@
 

FAQ: Help with this Definite Integral

What is a definite integral?

A definite integral is a mathematical concept used in calculus to calculate the area under a curve. It represents the accumulation of infinitely small rectangles under a curve, and is denoted by the integral symbol (∫).

How is a definite integral solved?

To solve a definite integral, you will need to use integration techniques such as u-substitution, integration by parts, or trigonometric substitution. Once you have simplified the integrand, you can use the fundamental theorem of calculus to evaluate the integral at its upper and lower bounds.

When do I need to use a definite integral?

A definite integral is used when you need to find the area under a curve, or the total amount of something that can be represented by a continuous function. It is also used to find the average value of a function over a given interval.

Are there any common mistakes when solving definite integrals?

Yes, some common mistakes when solving definite integrals include forgetting to include the integration constant, incorrectly applying integration techniques, and not properly setting up the integral limits. It is important to double check your work and be aware of these potential mistakes.

How can I check if my solution to a definite integral is correct?

You can check your solution by taking the derivative of the antiderivative you found and evaluating it at the integral limits. If the result matches the original integrand, then your solution is correct. You can also use online calculators or ask a math tutor for assistance in checking your work.

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