4. Set up, but do not evaluate, an integral that gives the surface area of the object obtained by rotating the parametric curve about the given axis. You may assume that the curve traces out exactly once for the given range of \(t\)’s.
Rotate \(x = 1 + \ln \left( {5 + {t^2}} \right)\hspace{0.25in}y = 2t - 2{t^2}\hspace{0.25in}0 \le t \le 2\) about the \(x\)-axis.
The first thing we’ll need here are the following two derivatives.
\[\frac{{dx}}{{dt}} = \frac{{2t}}{{5 + {t^2}}}\hspace{0.25in}\hspace{0.25in}\frac{{dy}}{{dt}} = 2 - 4t\] Show Step 2
We’ll need the \(ds\) for this problem.
\[ds = \sqrt {{{\left[ {\frac{{2t}}{{5 + {t^2}}}} \right]}^2} + {{\left[ {2 - 4t} \right]}^2}} \space dt = \sqrt {\frac{{4{t^2}}}{{{{\left( {5 + {t^2}} \right)}^2}}} + {{\left( {2 - 4t} \right)}^2}} \space dt\] Show Step 3
The integral for the surface area is then,
\[SA = \int_{{}}^{{}}{{2\pi y\space ds}} = {{\int_{0}^{2}{{2\pi \left( {2t - 2{t^2}} \right)\sqrt {\frac{{4{t^2}}}{{{{\left( {5 + {t^2}} \right)}^2}}} + {{\left( {2 - 4t} \right)}^2}} \space dt}}}}\]
Remember to be careful with the formula for the surface area! The formula used is dependent upon the axis we are rotating about.
Find the solution of the exponential equation \( 16 e ^ { x } = 12 \) in terms of logarithms, or correct to four decimal places. \( x = \)
Certain radioactive material decays in such a way that the mass remaining after \( t \) years is given by the function
\( m ( t ) = 100 e ^ { - 0.01 t } \)
where \( m ( t ) \) is measured in grams. (a) Find the mass at time \( t = 0 \) . Your answer is (b) How much of the mass remains after \( 20 \) years? Your answer is Round answers to \( 1 \) decimal place.
