Calculating the Volume of a Vase

Introduction

You are to determine the volume of the vase pictured here. The photograph on the right shows the construction of the vase and that its height is 50 cm.

Unlike practice problems from textbooks, most real life problems involve "messy" coefficients. Even if the function is easy to antidifferentiate, the arithmetic is usually tedious.

### In this lesson you will learn to...

• Capture the coordinates of points along the edge of the vase from its photograph
• Use those points and your calculator to determine a function
• Set up the definite integral for the volume of the vase (a solid of revolution) using your function (need help? disk method for calculating volume)
• Evaluate the integral using your calculator.

You may already know how to perform some of these. We have provided links to refresh your memory if needed.

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 Use the applet to find the coordinate values for the edge of the vase Your browser does not support Java, so nothing is displayed. Click anyplace on the image to move the cursor to that point. Notice the readout at the upper right gives the location of the cursor measured in pixels from the lower left corner of the image. You can fine tune the location of the cursor by clicking on the four arrows at the left, right, top, and bottom of the map. Each click moves the cursor one pixel in the direction indicated by the arrow. Click the blue button, labeled "Mark point," to record the coordinates of a point. Click it each time that you have moved the cursor to a new point whose coordinates you want to record. Click the red button, labeled "List points," to display the coordinates of the points whose coordinates you have recorded. A new window will appear with the information that you have recorded. If necessary, click the black button, labeled "Clear points," to clear the list of recorded points and begin again.

Now that you have coordinates from the applet,

• Set up the definite integral that represents the volume of the vase.
• Solve the integral using your calculator.