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Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.y = 8×2, y = 8

Solve the problem.A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N. How much work is done in stretching the spring 3 m from its equilibrium position?

Find the area of the surface generated when the given curve is revolved about the x-axis.  y =   on  

Use a calculator to approximate the area of the surface generated when the given curve is revolved about the x-axis. Round to two decimal places when necessary.y = x8  on [0, 1]

Solve the problem.A cylindrical water tank has height 10 m and radius 2 m (see figure). If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? Express the answer in terms of π. 

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines.y = 2x,y = x2; revolve about the y-axis

Write the integral that gives the surface area generated when the curve is revolved about the x-axis. Do not simplify.y = ln x4  on [1, ]

Find the volume of the described solid.The solid lies between planes perpendicular to the x-axis at  and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola  to the parabola .

“The authors and promoters of this desperate conspiracy have…meant only to amuse, by vague expressions of attachment to the parent state, and the strongest protestations of loyalty to me, whilst they were preparing for a general revolt… The resolutions of Parliament breathed a spirit of moderation and forbearance; conciliatory propositions accompanied the measures taken to enforce authority… I have acted with the same temper, anxious to prevent, if it had been possible…the calamities which are inseparable from a state of war; still hoping that my people in America would have discerned the traitorous views of their leaders, and have been convinced, that to be a subject of Great Britain, with all its consequences, is to be the freest member of any civil society in the known world.”-King George III, Speech to Parliament, October 27, 1775Based on the excerpt, which of the following best describes King George’s view of the colonists?

“We hold…that on their separation from the Crown of Great Britain, the several colonies became free and independent States, each enjoying the separate and independent right of self-government; and that no authority can be exercised over them…but by their consent… It is equally true, that the States…that the government created by it is a joint agency of the States, appointed to execute the powers enumerated and granted by that instrument; that all its acts not intentionally authorized are of themselves essentially null and void, and that the States have the right… to pronounce, in the last resort, authoritative judgement on the usurpations of the Federal Government… Such we deem to be inherent rights of the States.”-John C. Calhoun, statement adopted by a convention in South Carolina, 1832As described in the excerpt, which individual or body makes the final decision on whether a law is valid in a state?