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Chapter 5: Q. 51 (page 496)
The average value of the function on the interval.
The average value is.
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Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial as the quotient above the top line, and the polynomial 3x 鈭� 1 at the bottom as the remainder. Then
Explain why, if , then . Your explanation should include a discussion of domains and absolute values.
Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) An integral with which we could reasonably apply trigonometric substitution with .
(b) An integral with which we could reasonably apply trigonometric substitution with .
(c) An integral with which we could reasonably apply trigonometric substitution with .
Solve the integral:
Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution when we see , even though we can鈥檛 use the substitution unless the integrand involves the square root of? (Hint: Think about domains.)
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