Understanding the Sign of Integrals: Explaining x cos x without Evaluation

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In summary, the conversation discusses finding the integral of x cos x from 0 to pi/2 and from pi/2 to pi, and whether the integral of x cos x from 0 to pi would be positive, zero, or negative. The conversation also mentions using the graphs and the property that the integral of a positive function is positive. The speaker also expresses frustration with others not reading previous hints and looking for a complete solution.
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Natasha1
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I have been asked to explain without evaluating the integrals why the integral of x cos x from 0 to pi/2 is positive and the integral of x cos x from pi/2 to pi is negative. Also would I expect x cos x from 0 to pi to be positive, zero or negative? And why ?

How can I do this without evaluting it? :frown: some help would be much appreciated, thanks!
 
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  • #2
Do you know what the graphs look like? Keep in mind that a positive function times a negative function is negative, etc, and the integral of a positive function is positive, etc.
 
  • #3
If [tex]f(x)\geq 0[/tex] for x in [a,b], then [tex]\int_{a}^{b}f(x)dx \geq 0[/tex].
 
  • #4
Do not double-post, please.
Haven't you got enough information on this page (you posted it 3 days ago remember?)? If you have any further questions, why don't consider to post it there, instead of starting a brand new thread?
This proves that some never read the hints in the posts others have given, and think about it, instead, they are looking for a complete solution!
 
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FAQ: Understanding the Sign of Integrals: Explaining x cos x without Evaluation

What is the sign of an integral?

The sign of an integral refers to whether the value of the integral is positive or negative. It is determined by the area under the curve of the function being integrated.

Why is it important to understand the sign of integrals?

Understanding the sign of integrals is important because it allows us to determine the behavior of a function and make predictions about its values. It also helps us to evaluate integrals more efficiently.

How do you explain x cos x without evaluation?

To explain x cos x without evaluation, we can use the concept of symmetry. Since the cosine function is an even function, its graph is symmetrical about the y-axis. This means that the area under the curve from 0 to a positive value will be the same as the area from 0 to a negative value. Therefore, the integral of x cos x from 0 to a will always be 0.

Can the sign of an integral change?

Yes, the sign of an integral can change depending on the limits of integration and the shape of the function being integrated. For example, if the limits are changed from positive to negative, the sign of the integral will change.

How can understanding the sign of integrals help with applications?

Understanding the sign of integrals is important in many real-world applications, such as calculating work done by a force, finding the displacement of an object, and determining the area under a velocity-time graph. It also helps in solving optimization problems and predicting the behavior of a system.

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