91Ó°ÊÓ

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly twice on \((-\infty, \infty) .\) Explain how to use integration to find the area of the region bounded by the two curves.

Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line.

Fill in the blanks: A region \(R\) is revolved about the \(y\) -axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to __________ or using the shell method and integrating with respect to __________.

Given the velocity function \(v\) of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to \(x\) that gives the length of the following curves on the given interval. $$y=x^{3}+2 \text { on }[-2,5]$$

Why is the disk method a special case of the general slicing method?

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=x^{3} \text { on }[0,1]$$

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the semicircle \(y=\sqrt{1-x^{2}}\) and the \(x\) -axis, and whose cross sections through the solid perpendicular to the \(x\) -axis are squares.

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. The population of a town with a 2010 population of 90,000 grows at a rate of \(2.4 \% /\) yr. In what year will the population double its initial value (to \(180,000) ?\)

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. How long will it take an initial deposit of \(\$ 1500\) to increase in value to \(\$ 2500\) in a saving account with an APY of 3.1 \(\%\) ? Assume the interest rate remains constant and no additional deposits or withdrawals are made.