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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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Multiple Choice If \(\int _ { 3 } ^ { 7 } f ( x ) d x = 5\) and \(\int _ { 3 } ^ { 7 } g ( x ) d x = 3 ,\) then all of the following must be true except (A) $$\int _ { 3 } ^ { 7 } f ( x ) g ( x ) d x = 15$$ (B) $$\int _ { 3 } ^ { 7 } [ f ( x ) + g ( x ) ] d x = 8$$ (C) $$\int _ { 3 } ^ { 7 } 2 f ( x ) d x = 10$$ (D) $$\int _ { 3 } ^ { 7 } [ f ( x ) - g ( x ) ] d x = 2$$ (E) $$\int _ { 7 } ^ { 3 } [ g ( x ) - f ( x ) ] d x = 2$$
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { - 2 } ^ { 6 } 5 d x$$
In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{0}^{\pi / 2} \sin \left(x^{2}\right) d x$$
Use the inequality \(\sin x \leq x ,\) which holds for \(x \geq 0 ,\) to find an upper bound for the value of \(\int _ { 0 } ^ { 1 } \sin x d x . \)
Archimedes (287-212 B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times, discovered that the area under a parabolic arch like the one shown here is always two- thirds the base times the height. (a) Find the area under the parabolic arch \( y=6-x-x^2, -3 \leq x \leq 2 \) (b) Find the height of the arch. (c) Show that the area is two-thirds the base times the height.
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