91Ó°ÊÓ

give an expression that represents the statement. Do not simplify your expression. The future value at the end of 15 years of an income stream paying \(C\) dollars/year throughout the 15 years, at a \(2 \%\) interest rate compounded continuously.

Convert the Cartesian coordinates to polar coordinates. $$(1,1)$$

The region is rotated around the x-axis. Find the volume. $$\text { Bounded by } y=x^{2}, y=0, x=0, x=1$$

The density of cars (in cars per mile) down a 20 -mile stretch of the Pennsylvania Turnpike is approximated by $$\delta(x)=300(2+\sin (4 \sqrt{x+0.15}))$$, at a distance \(x\) miles from the Breezewood toll plaza. (a) Sketch a graph of this function for \(0 \leq x \leq 20\). (b) Write a Riemann sum that approximates the total number of cars on this 20 -mile stretch. (c) Find the total number of cars on the 20 -mile stretch.

(a) Find a Riemann sum which approximates the total mass of a \(3 \times 5\) rectangular sheet, whose density per unit area at a distance \(x\) from one of the sides of length 5 is \(1 /\left(1+x^{4}\right)\) (b) Calculate the mass.

The region is rotated around the x-axis. Find the volume. $$\text { Bounded by } y=(x+1)^{2}, y=0, x=1, x=2$$

Convert the Cartesian coordinates to polar coordinates. $$(-1,0)$$

The force, \(F,\) required to compress a spring by a distance \(x\) meters is given by \(F=3 x\) newtons. (a) Find the work done in compressing the spring from \(x=0\) to \(x=1\) and in compressing the spring from \(x=4\) to \(x=5\) (b) Which of the two answers is larger? Why?

Convert the Cartesian coordinates to polar coordinates. $$(\sqrt{6},-\sqrt{2})$$

Suppose that \(x\) measures the time (in hours) it takes for a student to complete an exam. All students are done within two hours and the density function for \(x\) is \(p(x)=\left\\{\begin{array}{ll}x^{3} / 4 & \text { if } 0