Problem $11.$ You are to construct an open rectangular box with a square base and a volume of $48c{m}^3.$ If material for the bottom costs $\frac{6}{c}{m}^2$ and material for the sides costs $\frac{4}{c}{m}^2,$ what dimensions will result in the least expensive box? What is the minimum cost? Support your answers with diagrams.

Problem $12.$ In a maximum power transfer theorem scenario, the power delivered to a load $P(R_L)$ is expressed as

$P(R_L)=\frac{V^2{R}_L}{(R+R_L{)}^2}$

where $V$ is the source voltage and $R$ is the internal resistance. Determine the load resistance $R_L$ that maximizes the power delivered to the load.

Problem $13.$ Use lower and lower sums with four rectangles to estimate the area under the graph of the functions.

$($a$)f(x)=x^2$ between $x=0$ and $x=1$

$($c$)f(x)=\frac{1}{x}$ between $x=1$ and $x=5$

$($b$)f(x)=x^3$ between $x=0$ and $x=1$

$($d$)f(x)=4-x^2$ between $x=-2$ and $x=2$

Problem $14.$ Use known area formulas to evaluate the integrals

$($a$){\displaystyle \underset{-3}{\overset{3}{}}}\sqrt[\phantom{2}]{9-x^2}dx$

$($c$){\displaystyle \underset{0}{\overset{b}{}}}4xdx,b>0$

$($b$){\displaystyle \underset{-2}{\overset{1}{}}}|x|dx$

$($d$){\displaystyle \underset{-1}{\overset{1}{}}}(1+\sqrt[\phantom{2}]{1-x^2})dx$

Problem $15.$ Find the derivative $d\frac{y}{d}x$ if

$($a$)y={\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{x}}{}}}costdt$

$($e$)y={\displaystyle \underset{\sqrt[\phantom{2}]{x}}{\overset{0}{}}}sin(t^2)dt$

$($b$)y={\displaystyle \underset{0}{\overset{x^3}{}}}{t}^{-\frac{2}{3}}dt$

$($c$)y={\displaystyle \underset{0}{\overset{tanx}{}}}se{c}^{2}tdt$

$($f$)y=n{t}_2^{x^2}sin(t^3)dt$

$($d$)y={\displaystyle \underset{0}{\overset{x}{}}}\sqrt[\phantom{2}]{1+t^2}dt$

$($g$)y={\displaystyle \underset{0}{\overset{sinx}{}}}\frac{dt}{\sqrt[\phantom{2}]{1-t^2}},|x|<mfrac><mrow><mi></mi></mrow><mrow><mn>2</mn></mrow></mfrac>$

Problem $16.$ Find $f(4)$ if ${\displaystyle \underset{0}{\overset{x}{}}}f(t)dt=xcosx.$

Problem $17.$ Use integration by substitution to evaluate the integrals.

$($a$){\displaystyle \underset{-2}{\overset{2}{}}}\frac{2}{(x+3{)}^4}dx$

$($e$){\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{\frac{}{2}}}{}}}xcos{x}^{2}dx$

$($i$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{3-2x}dx$

$($b$){\displaystyle \underset{0}{\overset{1}{}}}(x^2+\sqrt[\phantom{2}]{x})dx$

$($f$){\displaystyle \underset{1}{\overset{{}^2}{}}}\frac{sin\sqrt[\phantom{2}]{x}}{\sqrt[\phantom{2}]{x}}dx$

$($j$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}se{c}^2(3x+2)dx$

$($c$){\displaystyle \underset{\frac{}{2}}{\overset{0}{}}}\frac{1+cos2x}{2}dx$

$($g$){\displaystyle \underset{2}{\overset{5}{}}}\frac{xdx}{\sqrt[\phantom{2}]{1+x^2}}$

$($k$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{x^2}cos(\frac{1}{x}-1)dx$

$($d$){\displaystyle \underset{-\frac{}{3}}{\overset{-\frac{}{4}}{}}}(4se{c}^{2}x+\frac{}{x^2})dx$

$($h$){\displaystyle \underset{-4}{\overset{4}{}}}|x|dx$

$($l$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{x^2}\sqrt[\phantom{2}]{2-\frac{1}{x}}dx$