Chapter 5: Problem 29
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { e } \frac { 1 } { x } d x$$
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Writing to Learn In Example 2 (before rounding) we found the average temperature to be 65.17 degrees when we used the integral approximation, yet the average of the 13 discrete temperatures is only 64.69 degrees. Considering the shape of the temperature curve, explain why you would expect the average of the 13 discrete temperatures to be less than the average value of the temperature function on the entire interval.
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{\pi / 3} 4 \sec x \tan x d x$$
In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{\pi} \sin x d x$$
In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{0}^{1} \sqrt{1+x^{4}} d x$$
Multiple Choice At \(x=\pi,\) the linearization of \(f(x)=\int_{\pi}^{x} \cos ^{3} t d t\) (A) \(y=-1\) (B) \(y=-x \quad\) (C) \(y=\pi\) (D) \(y=x-\pi \quad\) (E) \(y=\pi-x\)
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