Chapter 5: Problem 25
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { - 2 } ^ { 6 } 5 d x$$
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Multiple Choice If \(\int _ { 2 } ^ { 5 } f ( x ) d x = 12\) and \(\int _ { 5 } ^ { 8 } f ( x ) d x = 4\) then all of the following must be true except (A) $$\int _ { 2 } ^ { 8 } f ( x ) d x = 16$$ (B) $$\int _ { 2 } ^ { 5 } f ( x ) d x - \int _ { 5 } ^ { 8 } 3 f ( x ) d x = 0$$ (C) $$\int _ { 5 } ^ { 2 } f ( x ) d x = - 12$$ (D) $$\int _ { - 5 } ^ { - 8 } f ( x ) d x = - 4$$ (E) $$\int _ { 2 } ^ { 6 } f ( x ) d x + \int _ { 6 } ^ { 8 } f ( x ) d x = 16$$
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{\pi / 6}^{5 \pi / 6} \csc ^{2} \theta d \theta$$
In Exercises \(41-44\) , find the total area of the region between the curve and the \(x\) -axis. $$y=x^{3}-4 x, \quad-2 \leq x \leq 2$$
In Exercises \(31 - 36 ,\) find the average value of the function on the interval, using antiderivatives to compute the integral. $$y = 3 x ^ { 2 } + 2 x , [ - 1,2 ]$$
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.) $$\int_{0}^{4} \sqrt{x} d x$$
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