Assignment $($Week $13)$: Applications of Integration

Instruction: Answer all the questions.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y=4.$

$($a$)y=x,y=3,x=0$

$($b$)y=\frac{2}{1+x},y=0,x=0,x=4$

Use the cylindrical shell method to find the volume of the solid generated when the region enclosed by the given curves is revolved about the $y-$axis.

$($a$)y=cos(x^2),x=0,x=\frac{1}{2}\sqrt[\phantom{2}]{},y=0$

$($b$)y=2x-1,y=-2x+3,x=2$

$($c$)y=e^{x^2},x=1,x=\sqrt[\phantom{2}]{3},y=0$

Find the arc length of the graph of the function over the indicated interval.

$($a$)y=ln(sinx),[\frac{}{4},\frac{3}{4}]$

$($b$)y=\frac{1}{2}(e^x+e^{-x}),[0,2]$

$($c$)x=\frac{1}{3}(y^2+2{)}^{\frac{3}{2}},0y4$

Use Hooke's Law to determine the work done by the variable force in the spring problem.

$($a$)$ A force of $5$ pounds compresses a $15-$inch spring a total of $3$ inches. How much work is done in compressing the spring $7$ inches?

$($b$)$ Eighteen foot$-$pounds of work is required to stretch a spring $4$ inches from its natural length. Find the work required to stretch the spring an additional $3$ inches.

It follows from Newton's Law of Universal Gravitation that the gravitational force exerted by the Earth on an object above the Earth's surface varies inversely as the square of its distance from the Earth's center. Thus, an object's weight $w(x)$ is related to its distance $x$ from the Earth's center by a formula of the form

$w(x)=\frac{k}{x^2}$

where $k$ is a constant of proportionality that depends on the mass of the object. The Earth is assumed to be a sphere of radius $4000$ mi $.$

$($a$)$ Find a formula for the weight $w(x)$ of a satellite that is $x$ mi from the Earth's surface if its weight on Earth is $6000$ lb $.$

$($b$)$ How much work is required to lift the satellite from the surface of the Earth to an orbital position that is $1000$ mi high?