Problem 28
In Exercises \(23-28,\) use areas to evaluate the integral. $$\int_{a}^{\sqrt{3} a} x d x, \quad a>0$$
Problem 28
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { e } \frac { 1 } { x } d x$$
Problem 29
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { e } \frac { 1 } { x } d x$$
Problem 29
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x$$
Problem 29
In Exercises \(29-32,\) express the desired quantity as a definite integral and evaluate the integral using Theorem \(2 .\) Find the distance traveled by a train moving at 87 mph from \(8 : 00\) A.M. to \(11 : 00\) A.M.
Problem 30
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { 4 } - x ^ { - 2 } d x$$
Problem 30
In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. \(\int_{0}^{5} x^{3 / 2} d x\)
Problem 31
In Exercises \(31 - 36 ,\) find the average value of the function on the interval, using antiderivatives to compute the integral. $$y = \sin x , \quad [ 0 , \pi ]$$
Problem 31
In Exercises \(29-32,\) express the desired quantity as a definite integral and evaluate the integral using Theorem \(2 .\) Find the calories burned by a walker burning 300 calories per hour between \(6 : 00\) P.M. and \(7 : 30\) P.M.
Problem 31
True or False If \(f\) is a positive, continuous, increasing function on \([a, b],\) then LRAM gives an area estimate that is less than the true area under the curve. Justify your answer.
