Problem 10
Integrals of Nonpositive Functions Show that if \(f\) is integrable then See page \(293 .\) \(f ( x ) \leq 0\) on \([ a , b ] \Rightarrow \int _ { 0 } ^ { b } f ( x ) d x \leq 0\)
Problem 10
In Exercises \(7-12,\) evaluate the integral. $$\int_{-4}^{-1} \frac{\pi}{2} d \theta$$
Problem 10
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int^{x^{2}} \cot 3 t d t$$
Problem 11
In Exercises \(7-12,\) evaluate the integral. $$\int_{-2.1}^{3.4} 0.5 d s$$
Problem 11
In Exercises \(11 - 14 ,\) use NINT to find the average value of the function on the interval. At what point (s) in the interval does the function assume its average value? $$y = x ^ { 2 } - 1 , [ 0 , \sqrt { 3 } ]$$
Problem 11
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{2}^{5 x} \frac{\sqrt{1+u^{2}}}{u} d u$$
Problem 11
In Exercises \(9-12,\) use RAM to estimate the area of the region enclosed between the graph of \(f\) and the \(x\) -axis for \(a \leq x \leq b\) . $$f(x)=e^{-x^{2}}, \quad a=0, \quad b=2$$
Problem 12
In Exercises \(9-12,\) use RAM to estimate the area of the region enclosed between the graph of \(f\) and the \(x\) -axis for \(a \leq x \leq b\) . $$f(x)=\sin x, \quad a=0, \quad b=\pi$$
Problem 12
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{\pi}^{\pi-x} \frac{1+\sin ^{2} u}{1+\cos ^{2} u} d u$$
Problem 12
In Exercises \(11 - 14 ,\) use NINT to find the average value of the function on the interval. At what point (s) in the interval does the function assume its average value? $$y = - \frac { x ^ { 2 } } { 2 } , \quad [ 0,3 ]$$
