Exam 9: Further Applications of the Integral and Taylor Polynomials

Calculate the Maclaurin polynomial for .

Find the surface area of the ellipsoid obtained by rotating the ellipse about the x-axis.

Approximate the arc length of the curve over the interval using the Trapezoidal Rule .

Calculate the Maclaurin polynomial for .

The plate shown in the figure, enclosed by the curves , , and , is submerged vertically in a fluid with density so that its top is at a depth of . Find the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Let . What is the coefficient of in the sixth Maclaurin polynomial for ?

Find the centroid of the shaded region in the figure below.

Five particles of equal mass are located at and . Find the center of mass of the system.

The face of a dam is an isosceles trapezoid of height 30 ft and bases 10 ft and 50 ft. The dam is oriented vertically with the larger base at the bottom. Find the force on the dam if the water level is 4 ft below the top of the dam. (The density of the water is .)

Let . A) Write the Maclaurin polynomial for . B) Use Taylor's Theorem for to write an integration formula for

Use implicit differentiation to compute the surface area of revolution about the x-axis of the part of the astroid in the first quadrant.

Compute the surface area of revolution of about the x-axis over the interval .

Use additivity of moments to find the center of mass of the region consisting of a semicircle on top of an isosceles trapezoid of height 1 and bases 2 and 4, as shown below.

Evaluate .

A thin plate shown in the following figure is bounded by the graphs of and . The plate is submerged in a fluid with density so that its top is level with the surface of the fluid. Calculate the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

A water tank is a cylinder 4 ft in diameter, standing on its base. If water fills the tank to a depth of 3 ft, what is the magnitude of the force exerted on the side of the tank? (The density of the water is .)

In the following figure, the centroid of the region enclosed by the graphs of , and lies on:

An infinite plate shown in the figure below, bounded by the graphs of and and the two axes, is submerged vertically in a fluid with density in kilograms per cubic meter; its top is level with the fluid surface. Calculate the fluid force on a side of the plate and write the answer in terms of and .

An infinite plate shown in the figure below, bounded by the graphs of , , and the y-axis, is submerged vertically in water, with its top ft below the water surface. Calculate the fluid force on a side of the plate. (The density of water is )

Calculate the Taylor polynomial for about .