Chapter 5: Problem 31
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{1}^{32} x^{-6 / 5} d x$$
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In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{2} x d x$$
In Exercises \(15-18,\) find the average value of the function on the interval without integrating, by appealing to the geometry of the region between the graph and the \(x\) -axis. $$f ( \theta ) = \tan \theta , \quad \left[ - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right]$$
Multiple Choice The area of the region enclosed between the graph of \(y=\sqrt{1-x^{4}}\) and the \(x\) -axis is $\mathrm (A) 0.886 (B) 1.253 (C) 1.414 (D) 1.571 (E) 1.748
Writing to Learn If \(a v ( f )\) really is a typical value of the integrable function \(f ( x )\) on \([ a , b ]\) , then the number \(a v ( f )\) should have the same integral over \([ a , b ]\) that \(f\) does. Does it? That is, does \(\int _ { a } ^ { b } a v ( f ) d x = \int _ { a } ^ { b } f ( x ) d x ?\) Give reasons for your answer.
Writing to Learn A dam released 1000\(\mathrm { m } ^ { 3 }\) of water at 10\(\mathrm { m } ^ { 3 / \mathrm { min } }\) and then released another 1000\(\mathrm { m } ^ { 3 }\) at 20\(\mathrm { m } ^ { 3 / \mathrm { min } }\) . What was the average rate at which the water was released? Give reasons for your answer.
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