Given the region bounded by the graphs of y = x sin x, y = 0, x = 0, and x = π, find (a) the area of the region. (b) the volume of the solid generated by revolving the region about the x-axis. (c) the volume of the solid generated by revolving the region about the y-axis. (d) the centroid of the region.

Use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. $y=x \sin x, y=0, x=\pi$

Find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=x$ about the $x$-axis.

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve

$$y = e ^ { x }$$

, and the line x = ln 2 about the line x = ln 2.

The annual sales $a_n$ (in millions of dollars) for Avon Products, Inc. from $1993$ through $2002$ are given below as ordered pairs of the form $\left(n, a_n\right)$, where $n$ represents the year, with $n=3$ corresponding to $1993$. (Source: $2002$ Avon Products, Inc. Annual Report)

$$\begin{aligned} &(3,3844),(4,4267),(5,4492),(6,4814),(7,5079),(8,5213), \\ &(9,5289),(10,5682),(11,5958),(12,6171) \end{aligned}$$

Use the regression capabilities of a graphing utility to find a model of the form

$$a_n=c e^{k n}, \quad n=3,4,5, \ldots, 12$$

for the data. Graphically compare the points and the model.

Let R be the region bounded by y=sin x and the x-axis on the interval [0, $\pi$]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis?

Find the volume of the solid generated by revolving the region about the y-axis. The region bounded above by the curve y = √x and below by the line y = x

Find the area bounded by the graphs of $y=x \sin x$ and $y=0$ over interval.

$$[\pi, 2 \pi]$$

Find the area of the region bounded by the graphs of the equations. y = sin x, y = sin³ x, x = 0, x = π/2

Find the volume of the solid generated by revolving the region about the given line. The region in the first quadrant bounded above by the line y = 2, below by the curve y = 2 sin x, 0 ≤ x ≤ π/2, and on the left by the y-axis, about the line y = 2.

A particle moves on a straight line with velocity function v(t) = sin wt cos^2 wt. Find its position function s = f(t) if f(0) = 0.

Roman saves $500 each year in an account earning interest at an annual rate of 4% compounded annually. How much interest will the account earn at the end of each of the first 3 years?

A solid is generated by revolving the region bounded by y = √9-x² and y = 0 about the axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole.

Find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=\sqrt{x}$ about the $x$-axis.