Calculus Simulation 2

YOU WILL LEARN ABOUT:

How to use the Disk Method to find the volume of a solid that is obtained by rotating an appropriate plane region about a line.

The formula for the volume of the solid of revolution that has disks as its cross-section is given by

, if the axis of rotation is the x-axis.
, if the axis of rotation is the y-axis.

Observe that R (x) and R (y) are the radii of the disks drawn at x and y respectively.
*Disk Method is an application of the method of slicing.

Professionals, like architects, Pharmaceutical companies , and engineers often need to calculate the volume of objects. Here are some examples.

Lathes are machines that are used to fabricate a diverse set of objects that have circular cross sections ranging from a piece of jewelry by craftsmen to a gun barrel or bullets by military engineers. Sometimes the volume of an object needs to be known before it can be shaped by the lathe. For instance, when designing a bullet that can fit into a certain size of the gun barrel we need to know its specific volume.

Pharmaceutical companies can determine the volume of each pill to find out the correct dosage and also the number of pills a bottle can hold based on its volume.

Symphony drums are hemispherical or parabolic in shape. The shape of the ball determines its tone. Calculating the volume would allow you to create drums of different pitches.

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

X Axis

Y Axis

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VOLUMES OF REVOLUTION - DISK METHOD

To rotate grid, click and drag

REVOLVE REGION

# OF DISKS

AXIS OF REVOLUTION

x y

Region 1

Region 2

Region 3

$\begin{matrix} Region 1 y = - \frac{10}{12} \sqrt{(12^{2} - x^{2})}, y = 0 \end{matrix}$

$\begin{matrix} Volume \int_{a}^{b} π y^{2} d x \approx 0 \end{matrix}$

$\begin{matrix} Volume \int_{c}^{d} π x^{2} d y \approx 0 \end{matrix}$

$\begin{array}{ll} Number of Disks : & 0 \\ Width of Disks : & 0.00 \\ Sum of Volume of Disks \approx & 0.00 units3 \end{array}$

Note

Note that the disk method is not applicable here since not all the slices of the solid generated are disks. However, you will be able to compute its volume using washer method which will be covered in the next module.

YOU WILL LEARN ABOUT:

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.

Select any of the regions from the three options given at the bottom left-hand corner of the screen. Rotate the region around either x or y axis to obtain the solid and answer the following questions.