Problem 40
Writing to Learn A driver averaged 30\(\mathrm { mph }\) on a 150 -mile trip and then returned over the same 150 miles at the rate of 50\(\mathrm { mph }\) . He figured that his average speed was 40\(\mathrm { mph }\) for the entire trip. (a) What was his total distance traveled? (b) What was his total time spent for the trip? (c) What was his average speed for the trip? (d) Explain the error in the driver's reasoning.
Problem 41
In Exercises \(41-44\) , find the total area of the region between the curve and the \(x\) -axis. $$y=2-x, \quad 0 \leq x \leq 3$$
Problem 41
True or False If \(\int_{a}^{b} f(x) d x>0,\) then \(f(x)\) is positive for all \(x\) in \([a, b] .\) Justify your answer.
Problem 41
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.
Problem 42
Use the inequality \(\sin x \leq x ,\) which holds for \(x \geq 0 ,\) to find an upper bound for the value of \(\int _ { 0 } ^ { 1 } \sin x d x . \)
Problem 42
In Exercises \(41-44\) , find the total area of the region between the curve and the \(x\) -axis. $$y=3 x^{2}-3, \quad-2 \leq x \leq 2$$
Problem 42
True or False If \(f(x)\) is positive for all \(x\) in \([a, b],\) then \(\int_{a}^{b} f(x) d x>0 .\) Justify your answer.
Problem 43
The inequality sec \(x \geq 1 + \left( x ^ { 2 } / 2 \right)\) holds on \(( - \pi / 2 , \pi / 2 ) .\) Use it to find a lower bound for the value of \(\int _ { 0 } ^ { 1 } \sec x d x .\)
Problem 43
In Exercises \(41-44\) , find the total area of the region between the curve and the \(x\) -axis. $$y=x^{3}-3 x^{2}+2 x, \quad 0 \leq x \leq 2$$
Problem 43
Multiple Choice If \(\int_{2}^{5} f(x) d x=18,\) then \(\int_{2}^{5}(f(x)+4) d x=\) \((\mathbf{A}) 20 \quad(\mathbf{B}) 22\) \((\mathrm{C}) 23 \quad(\mathrm{D}) 25 \quad(\mathrm{E}) 30\)
