91Ó°ÊÓ

\(31-36=\) Evaluate the integral by interpreting it in terms of areas. $$\int_{0}^{9}\left(\frac{1}{3} x-2\right) d x$$

\(31-36=\) Evaluate the integral by interpreting it in terms of areas. $$\int_{-3}^{0}\left(1+\sqrt{9-x^{2}}\right) d x$$

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate \(f=f(t)\) , where \(t\) is the time measured in months since its last over- haul. Because a fixed cost \(A\) is incurred each time the machine is overhauled, the company wants to determine the optimal time \(T\) (in months) between overhauls. (a) Explain why \(\int_{0}^{t} f(s) d s\) represents the loss in value of the machine over the period of time \(t\) since the last overhaul. (b) Let \(C=C(t)\) be given by $$ C(t)=\frac{1}{t}\left[A+\int_{0}^{t} f(s) d s\right] $$ What does \(C\) represent and why would the company want to minimize \(C\) ? (c) Show that \(C\) has a minimum value at the numbers \(t=T\) where \(C(T)=f(T)\)

Evaluate the indefinite integral. $$\int x(2 x+5)^{8} d x$$

\(33-34=\) Calculate the area of the region that lies under the curve and above the \(x\) -axis. $$y=2 x-x^{2}$$

\(31-36=\) Evaluate the integral by interpreting it in terms of areas. $$\int_{-5}^{5}\left(x-\sqrt{25-x^{2}}\right) d x$$

Evaluate the indefinite integral. $$\int \frac{x^{3}}{\sqrt{x^{2}+1}} d x$$

\(35-36=\) Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. $$y=\sin x, 0 \leqslant x \leqslant \pi$$

\(31-36=\) Evaluate the integral by interpreting it in terms of areas. $$\int_{-1}^{2}|x| d x$$

Evaluate the indefinite integral. $$\int \frac{1+x}{1+x^{2}} d x$$