Chapter 5: Problem 28
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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In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{-1}^{1}(r+1)^{2} d r$$
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{\pi / 4}^{3 \pi / 4} \csc x \cot x d x$$
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 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. $$\int_{0}^{2} x d x$$
Show that if \(F ^ { \prime } ( x ) = G ^ { \prime } ( x )\) on \([ a , b ] ,\) then \(F ( b ) - F ( a ) = G ( b ) - G ( a )\)
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