Chapter 7

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=\cos (x) \quad I=[0, \pi / 2] $$

A steam shovel lifts a 500 pound load of gravel from the ground to a point 80 feet above the ground. The gravel is fine, however, and it leaks from the shovel at the rate of 1 pound per second. If it takes the steam shovel one minute to lift its load at a constant rate, then how much work is performed?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=x y, y(x)=C e^{x^{2} / 2} $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the triangular region with vertices \((0,0),(0,2),\) and (6,0)\(;\) about \(x=3\)

In each of Exercises \(1-4,\) the graph of the given function \(f\) with given domain \(I\) is a line segment. Use formula (7.2.3) to calculate the arc length of the graph of \(f\). Verify that this length is the distance between the two endpoints. $$ f(x)=3 x \quad I=[1,4] $$

In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=\sqrt{x},\) above the \(x\) -axis, and between \(x=1\) and \(x=3\).

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=x^{2} \quad I=[3,7] $$

On the surface of the earth, a rocket weighs 10,000 newtons. How much work is performed lifting the rocket to a height 100 kilometers above the surface of the earth, assuming that the direction of the force, as well as the motion, is straight up?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=2 x y^{2}, y(x)=1 /\left(C-x^{2}\right) $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the triangular region with vertices \((0,0),(0,2),\) and (6,0)\(;\) about \(x=-1\)

In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region between the \(x\) -axis and the parabola \(y=\) \(4-x^{2}\) for \(-2 \leq x \leq 2\)

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the triangular region with vertices \((0,0),(0,2),\) and (6,0)\(;\) about \(x=2\).

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=1 / x \quad I=[1,4] $$

An object that weighs \(W_{0}\) at the surface of the earth weighs $$ 15697444 \cdot \frac{W_{0}}{(3962+y)^{2}} $$ when it is \(y\) miles above the surface of the earth. How much work is done lifting an object from the surface of the earth, where it weighs 100 pounds, straight up to a height 100 miles above Earth?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=x-3 y, y(x)=x / 3-1 / 9+C e^{-3 x} $$

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=3 x^{2}-6 x+1 \quad I=[3,7] $$

When a mass \(M\) measured in slugs is \(y\) miles above the surface of the earth, its weight in pounds is $$ 502318208 \cdot \frac{M}{(3962+y)^{2}} $$ How much work is done lifting a satellite from the surface of the earth, where it weighs 800 pounds, to an orbit 200 miles above Earth?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=e^{x}+y, y(x)=e^{x}(x+C) $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the triangular region with vertices \((0,0),(0,2),\) and (6,0)\(;\) about \(x=2\)

The graph of the given function \(f\) with given domain \(I\) is a line segment. Use formula (7.2.3) to calculate the arc length of the graph of \(f\). Verify that this length is the distance between the two endpoints. $$ f(x)=10-2 x \quad I=[0,5] $$

In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region above the \(x\) -axis, below the graph of \(y=\) \(\sec (x),\) to the right of \(x=0,\) and to the left of \(x=\pi / 4\)

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=\sin (x) \quad I=[\pi / 3, \pi] $$

To lift an object straight up from the surface of the earth to a height \(25 \mathrm{~km}\) above the surface of the earth requires \(58610091 \mathrm{~J}\) of work. What is the mass of the object?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=x+y, y(x)=C e^{x}-x-1 $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the first quadrant region bounded by \(y=4 x-x^{3}\) and the \(x\) -axis; about \(x=1\).

In each of Exercises \(5-12,\) calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=2+x^{3 / 2} \quad I=[1,4] $$

In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region above the \(x\) -axis, below the graph of \(y=\) \(\exp (x)\), to the right of \(x=0\), and to the left of \(x=\ln (2)\).

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=1+\cos (x) \quad I=[0,2 \pi] $$

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=x+x y, y(x)=C e^{x^{2} / 2}-1 $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the first quadrant region bounded above by \(y=x-x^{2}\) and the \(x\) -axis; about \(x=0\).

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=(x-1)^{3 / 2} \quad I=[1,5] $$

In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=x^{1 / 3}\), above the \(x\) -axis, and between \(x=1\) and \(x=8\).

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=(x-1)^{1 / 2} \quad I=[2,5] $$

A rocket climbs straight up. Its initial total weight, including fuel, is 7000 pounds. Fuel is consumed at the constant rate of 30 pounds per mile. Taking into account the decrease in weight due to fuel consumption but disregarding the decrease in weight due to increasing elevation, approximate the work done in lifting the rocket the first 20 miles into space.

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=y+x^{2}, y(x)=C e^{x}-x^{2}-2 x-2 $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the region bounded above by \(y=1 / x\), below by the \(x\) axis, and on the sides by the vertical lines \(x=1\) and \(x=2\); about \(x=-3\)

In each of Exercises 7-12, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region in the first quadrant that is bounded on the left by the \(y\) -axis, on the right by the graph of \(y=\arcsin (x),\) and above by \(y=\pi / 2\).

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=3+(2 x+1)^{3 / 2} \quad I=[0,4] $$

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=x^{1 / 2}-x^{1 / 3} \quad I=[0,64] $$

A man stands at the top of a tall building and pulls a chain up the side of the building. The chain is 50 feet long and weighs 3 pounds per linear foot. How much work does the man do in pulling the chain to the top?

Verify that the given function \(y\) satisfies the given differential equation. In each expression for \(y(x)\) the letter \(C\) denotes a constant. $$ \frac{d y}{d x}=\frac{2 x-y}{x+y}, y=-x+\sqrt{3 x^{2}+2 C} $$

Find the moment of the given region \(\mathcal{R}\) about the given vertical axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the first quadrant region bounded above by \(y=\frac{\sin (x)}{x}\) below by the \(x\) -axis, and on the sides by \(x=\pi / 6\) and \(x=\pi / 2\); about \(x=0\).

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=(5-2 x)^{3 / 2} \quad I=[1 / 2,2] $$

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=60 / x^{2} \quad I=[1,3] $$

Solve the given differential equation. $$ \frac{d y}{d x}=6 \sqrt{x y} $$

Find the moment of the given region \(\mathcal{R}\) about the \(x\) -axis. Assume that \(\mathcal{R}\) has uniform unit mass density. \(\mathcal{R}\) is the triangular region with vertices \((0,0),(0,2),\) and (6,0).

In each of Exercises 7-12, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region to the right of the \(y\) -axis, to the left of the curve \(y=\ln (x)\), above the \(x\) -axis, and below \(y=1\).

Calculate the arc length \(L\) of the graph of the given function over the given interval. $$ f(x)=\ln (\cos (x)) \quad I=[0, \pi / 3] $$

In each of Exercises \(1-12,\) calculate the average value of the given function on the given interval. $$ f(x)=e^{x} \quad I=[-1,1] $$

Solve the given differential equation. $$ \left(4+y^{2}\right) \frac{d y}{d x}=x^{2} $$