Chapter 8

Find the center of mass of the three particles having masses of 1,2, and 3 slugs and located at the points \((-1,3),(2,1)\), and \((3,-1)\), respectively.

A spring has a natural length of 8 in. If a force of \(20 \mathrm{lb}\) stretches the spring \(\frac{1}{2}\) in., find the work done in stretching the spring from \(8 \mathrm{in}\). to \(11 \mathrm{in}\).

The base of a solid is a circle having a radius of \(r\) units. Find the volume of the solid if all plane sections perpendicular to a fixed diameter of the base are equilateral triangles.

x^{2}=-y ; y=-4

Find the center of mass of the four particles having masses of \(2,3,3\), and 4 slugs and located at the points \((-1,-2)\), \((1,3),(0,5)\), and \((2,1)\), respectively.

A square plate of side \(4 \mathrm{ft}\) is submerged vertically in a tank of water and its center is \(2 \mathrm{ft}\) below the surface. Find the force due to liquid pressure on one side of the plate.

A spring has a natural length of 10 in. , and a 30 -lb force stretches it to \(11 \frac{1}{2}\) in. Find the work done in stretching the spring from 10 in. to 12 in. Then find the work done in stretching the spring from 12 in. to 14 in.

The base of a solid is a circle with a radius of \(r\) units, and all plane sections perpendicular to a fixed diameter of the base are isosceles right triangles having the hypotenuse in the plane of the base. Find the volume of the solid.

y^{2}=-x ; x=-2 ; x=-4

Prove that the centroid of three particles, having equal masses, in a plane lies at the point of intersection of the medians of the triangle having as vertices the points at which the particles are located.

A spring has a natural length of 6 in. A \(12,000-\mathrm{lb}\) force compresses the spring to \(5 \frac{1}{2}\) in. Find the work done in compressing it from 6 in. to 5 in. Hooke's law holds for compression as well as for extension.

A plate in the shape of an isosceles right triangle is submerged vertically in a tank of water, with one leg lying in the surface. The legs are each \(6 \mathrm{ft}\) long. Find the force due to liquid pressure on one side of the plate.

Find the volume of a right pyramid having a height of \(h\) units and a square base of side \(a\) units.

Find the volume of the tetrahedron having 3 mutually perpendicular faces and three mutually perpendicular edges whose lengths have measures \(a, b\), and \(c\).

\(x^{3}=2 y^{2} ; x=0, y=-2\)

The base of a solid is a circle with a radius of 4 in. , and each plane section perpendicular to a fixed diameter of the base is an isosceles triangle having an altitude of 10 in. and a chord of the circle as a base. Find the volume of the solid.

The region bounded by \(y=x^{3}, x=2\), and the \(x\) axis, about the line \(x=2\). Take the rectangular elements perpendicular to the axis of revolution.

The face of a dam adjacent to the water is vertical, and its shape is in the form of an isosceles triangle \(250 \mathrm{ft}\) wide across the top and \(100 \mathrm{ft}\) high in the center. If the water is \(10 \mathrm{ft}\) deep in the center, find the total force on the dam due to liquid pressure.

The base of a solid is a circle with a radius of 9 in., and each plane section perpendicular to a fixed diameter of the base is a square having a chord of the circle as a diagonal. Find the volume of the solid.

An oil tank is in the shape of a right-circular cylinder \(4 \mathrm{ft}\) in diameter, and its axis is horizontal. If the tank is half full of oil weighing \(50 \mathrm{lb} / \mathrm{ft}^{3}\), find the total force on one end due to liquid pressure.

A right-circular cylindrical tank with a depth of \(12 \mathrm{ft}\) and a radius of \(4 \mathrm{ft}\) is half full of oil weighing \(60 \mathrm{lb} / \mathrm{ft}^{3}\). Find the work done in pumping the oil to a height \(6 \mathrm{ft}\) above the tank.

Two right-circular cylinders, each having a radius of \(r\) units, have axes that intersect at right angles. Find the volume of the solid common to the two cylinders.

The face of the gate of a dam is in the shape of an isosceles triangle \(4 \mathrm{ft}\) wide at the top and \(3 \mathrm{ft}\) high. If the upper edge of the face of the gate is \(15 \mathrm{ft}\) below the surface of the water, find the total force due to liquid pressure on the gate.

A cable \(200 \mathrm{ft}\) long and weighing \(4 \mathrm{lb} / \mathrm{ft}\) is hanging vertically down a well. If a weight of \(100 \mathrm{lb}\) is suspended from the lower end of the cable, find the work done in pulling the cable and weight to the top of the well.

\text { Find the volume of the sphere generated by revolving the circle whose equation is } x^{2}+y^{2}=r^{2} \text { about a diameter. }

\(y^{2}=x-1 ; x=3\)

The face of a gate of a dam is vertical and in the shape of an isosceles trapezoid \(3 \mathrm{ft}\) wide at the top, \(4 \mathrm{ft}\) wide at the bottom, and \(3 \mathrm{ft}\) high. If the upper base is \(20 \mathrm{ft}\) below the surface of the water, find the total force due to liquid pressure on the gate.

A bucket weighing \(20 \mathrm{lb}\) containing \(60 \mathrm{lb}\) of sand is attached to the lower end of a \(100 \mathrm{ft}\) long chain that weighs \(10 \mathrm{lb}\) and is hanging in a deep well. Find the work done in raising the bucket to the top of the well.

A wedge is cut from a solid in the shape of a right-circular cone having a base radius of \(5 \mathrm{ft}\) and an altitude of \(20 \mathrm{ft}\) by two half planes through the axis of the cone. The angle between the two planes has a measurement of \(30^{\circ}\). Find the volume of the wedge cut out.

Find by integration the volume of a right-circular cone of altitude \(h\) units and base radius \(a\) units.

The length of a rod is \(L \mathrm{ft}\) and the center of mass of the rod is at the point \(\frac{3}{4} L \mathrm{ft}\) from the left end. If the measure of the linear density at a point is proportional to a power of the measure of the distance of the point from the left end and the linear density at the right end is 20 slugs/ft, find the linear density at a point \(x \mathrm{ft}\) from the left end. Assume the mass is measured in slugs.

The face of a dam adjacent to the water is inclined at an angle of \(30^{\circ}\) from the vertical. The shape of the face is a rectangle of width \(50 \mathrm{ft}\) and slant height \(30 \mathrm{ft}\). If the dam is full of water, find the total force due to liquid pressure on the face.

Find the volume of the solid generated by revolving about the \(x\) axis the region bounded by the curve \(y=x^{3}\) and the lines \(y=0\) and \(x=2\)

\(y=\sqrt{x} ; y=x^{3}\)

Prove that the distance from the centroid of a triangle to any side of the triangle is equal to one-third the length of the altitude to that side.

The total mass of a rod of length \(L \mathrm{ft}\) is \(M\) slugs and the measure of the linear density at a point \(x \mathrm{ft}\) from the left end is proportional to the measure of the distance of the point from the right end. Show that the linear density at a point on the rod \(x \mathrm{ft}\) from the left end is \(2 M(L-x) / L^{2}\) slugs \(/ \mathrm{ft}\).

If the centroid of the region bounded by the parabola \(y^{2}=4 p x\) and the line \(x=a\) is to be at the point \((p, 0)\), find the value of \(a\).

The bottom of a swimming pool is an inclined plane. The pool is \(2 \mathrm{ft}\) deep at one end and \(8 \mathrm{ft}\) deep at the other. If the width of the pool is \(25 \mathrm{ft}\) and the length is \(40 \mathrm{ft}\), find the total force due to liquid pressure on the bottom.

A tank in the form of a rectangular parallelepiped 6 ft deep, 4 ft wide, and \(12 \mathrm{ft}\) long is full of oil weighing \(50 \mathrm{lb} / \mathrm{ft}^{3} .\) When one-third of the work necessary to pump the oil to the top of the tank has been done, find by how much the surface of the oil is lowered.

Find the volume of the solid generated by revolving about the line \(x=-4\) the region bounded by that line and the parabola \(x=4+6 y-2 y^{2}\).

\(y^{3}=x^{2} ; x-3 y+4=0\)

Find the volume of the solid generated by revolving the region bounded by the curves \(y^{2}=4 x\) and \(y=x\) about the \(x\) axis.

A one horsepower motor can do \(550 \mathrm{ft}-\mathrm{lb}\) of work per second. If a \(0.1 \mathrm{hp}\) motor is used to pump water from a full tank in the shape of a rectangular parallelepiped \(2 \mathrm{ft}\) deep, \(2 \mathrm{ft}\) wide, and \(6 \mathrm{ft}\) long to a point \(5 \mathrm{ft}\) above the top of the tank, how long will it take?

A meteorite is \(a\) miles from the center of the earth and falls to the surface of the earth. The force of gravity is inversely proportional to the square of the distance of a body from the center of the earth. Find the work done by gravity if the weight of the meteorite is \(w \mathrm{lb}\) at the surface of the earth. Let \(R\) miles be the radius of the earth.

An oil tank in the shape of a sphere has a diameter of \(60 \mathrm{ft}\). How much oil does the tank contain if the depth of the oil is \(25 \mathrm{ft}\) ?

\(x=y^{2}-y ; x=y-y^{2}\)

Find the volume of the solid generated if the region bounded by the parabola \(y^{2}=4 a x(a>0)\) and the line \(x=a\) is revolved about \(x=a\).

Find the volume of the solid generated by revolving the region bounded by the curve \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) about the \(y\) axis.

Find the center of mass of the lamina bounded by the parabola \(2 y^{2}=18-3 x\) and the \(y\) axis if the area density at any point \((x, y)\) is \(\sqrt{6-x}\) slugs \(/ \mathrm{ft}^{2}\)

Suppose that a cylindrical hole with a radius of \(r\) units is bored through a solid wooden hemisphere of radius \(2 r\) units, so that the axis of the cylinder is the same as the axis of the hemisphere. Find the centroid of the solid remaining.