Finding value of the following Integrals
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Dr.Peterson
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Sure. What help do you need?
You got the first one right, if that helps. Each of these uses a property of definite integrals, and (a) uses the fact that when you interchange the limits of integration, you negate the integral.
What thoughts do you have on the others?
Dr.Peterson
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Yes, C is good; it uses the fact that the integral of a sum (over the same interval) is the sum of the integrals.
I'd like you to at least make a try at the others (even if only an observation about possible relationships), because you may have better ideas than you realize, and I want to be able to commend you!
The first thing I'd do with this problem is to examine the givens so I know the raw material I have to work with. Did you notice that the first two given integrals have the same integrand, with different intervals, [-5,-2] and [-2,5]? Did you notice that together these cover the interval [-5,5]? Likewise, I would have noticed that the third given integral covers the interval [-2,5], but backward and with a different function.
Then I'd look at each question. The first, as you've observed, is the third given integral reversed, so the relationship is easy. The second covers that interval [-5,5], so ...
Then (c) is the sum of the second and third given integrals, with that sign change.
But (d) is different, isn't it? Its interval is unrelated to the given ones. So I'd look more closely at the question itself. Might there be a relationship between these two integrals, rather than to a given one?
Now, (e) seems to leave us nothing to work with. If the problem said, "determine the values (if possible) ...", I'd answer "insufficient data". If it gave some additional information about function h, I might be able to say something different. Are you sure there's nothing you left out? Is it possible that there just isn't a solution, and this book always implicitly allows for such an answer? They may just want you to know the difference between rules and wild guesses, and be able to say when no rule applies.
I'd like you to at least make a try at the others (even if only an observation about possible relationships), because you may have better ideas than you realize, and I want to be able to commend you!
The first thing I'd do with this problem is to examine the givens so I know the raw material I have to work with. Did you notice that the first two given integrals have the same integrand, with different intervals, [-5,-2] and [-2,5]? Did you notice that together these cover the interval [-5,5]? Likewise, I would have noticed that the third given integral covers the interval [-2,5], but backward and with a different function.
Then I'd look at each question. The first, as you've observed, is the third given integral reversed, so the relationship is easy. The second covers that interval [-5,5], so ...
Then (c) is the sum of the second and third given integrals, with that sign change.
But (d) is different, isn't it? Its interval is unrelated to the given ones. So I'd look more closely at the question itself. Might there be a relationship between these two integrals, rather than to a given one?
Now, (e) seems to leave us nothing to work with. If the problem said, "determine the values (if possible) ...", I'd answer "insufficient data". If it gave some additional information about function h, I might be able to say something different. Are you sure there's nothing you left out? Is it possible that there just isn't a solution, and this book always implicitly allows for such an answer? They may just want you to know the difference between rules and wild guesses, and be able to say when no rule applies.
Dr.Peterson
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Yes, I agree.
Dr.Peterson
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Yes, that's what I agreed with.
