Volume of a Container A container can be modeled by...

Volume of a Container A container can be modeled by revolving the graph of $$y=\left\{\begin{array}{l}{\sqrt{0.1 x^{3}-2.2 x^{2}+10.9 x+22.2}} \\ {2.95}\end{array}\right.$$ $\begin{array}{l}{0 \leq x \leq 11.5} \\ {11.5< x \leq 15}\end{array}$ about the $x$ -axis, where $x$ and $y$ are measured in centimeters. Use a graphing utility to graph the function. Find the volume of the container analytically.

Volume of a Container A container can be modeled by
revolving the graph of
$$y=\left\{\begin{array}{l}{\sqrt{0.1 x^{3}-2.2 x^{2}+10.9 x+22.2}} \\ {2.95}\end{array}\right.$$
$\begin{array}{l}{0 \leq x \leq 11.5} \\ {11.5< x \leq 15}\end{array}$
about the $x$ -axis, where $x$ and $y$ are measured in centimeters.
Use a graphing utility to graph the function. Find the volume
of the container analytically.

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Key Concepts

Volume of Revolution

This concept involves finding the volume of a solid formed by rotating a plane region about an axis. It is a key application of integral calculus that computes the accumulated volume of infinitesimally thin slices (disks or washers) generated by the rotation of a curve or a region.

Disk Method

The disk method is a technique used to calculate the volume of a solid of revolution when the solid is generated by revolving a function about an axis. By considering each cross-sectional slice as a disk with a small thickness, the volume is found by integrating ? times the square of the radius across the interval of interest.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. When working with integrals of piecewise functions, it is important to split the integral into sections corresponding to each definition. This ensures that the correct expression is used for each part of the domain during integration.

Definite Integration

Definite integration is the process of calculating the integral of a function over a specific interval, yielding a numerical value. It is used to sum up the contributions of infinitesimally small parts (such as disks in the disk method) over a range, providing the total volume or area described by the function.

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