91Ó°ÊÓ

A function \(f\) is defined below. Use geometric formulas to find \(\int_{0}^{8} f(x) d x\) $$f(x)=\left\\{\begin{array}{ll}4, & x<4 \\ x, & x \geq 4\end{array}\right.$$

Give an example of a function that is integrable on the interval [-1,1] , but not continuous on [-1,1] .

Use \(a(t)=-32\) feet per second per second as the acceleration due to gravity. Show that the height above the ground of an object thrown upward from a point \(s_{0}\) feet above the ground with an initial velocity of \(v_{0}\) feet per second is given by the function $$ f(t)=-16 t^{2}+v_{0} t+s_{0} $$.

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let \(n=4\) and round your answers to four decimal places. Use a graphing utility to verify your result. $$ \int_{2}^{6} \ln x d x $$

Find the indefinite integral using the formulas of Theorem 4.24 \(\int \frac{d x}{(x+2) \sqrt{x^{2}+4 x+8}}\)

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} t \cos t d t $$

Find \(F^{\prime}(x)\). $$ F(x)=\int_{-x}^{x} t^{3} d t $$

The sales \(S\) (in thousands of units) of a seasonal product are given by the model \(S=74.50+43.75 \sin \frac{\pi t}{6}\) where \(t\) is the time in months, with \(t=1\) corresponding to January. Find the average sales for each time period. (a) The first quarter \((0 \leq t \leq 3)\) (b) The second quarter \((3 \leq t \leq 6)\) (c) The entire year \((0 \leq t \leq 12)\)