91Ó°ÊÓ

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. $$ f(x)=3 x \quad I=[1,4] $$

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

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?

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

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 \(x=3\).

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

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 \(-2 \leq x \leq 2\)

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?

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