Dynamic Calculus
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The volume of a solid of revolution (y-axis)
What is the volume of the solid of revolution generated when the area bounded by \(y = {x^2}\), the \(y\)-axis and the lines \(y = a\) and \(y = b\) is rotated about the \(y\)-axis?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Calculate the area bounded by the curve, the \(y\)-axis and the abscissae \(y = 2\) and \(y = 16\).
- Rotate the region in question (1) about the \(y\)-axis by moving the 'slider' to the right. Using calculus, calculate the volume of the solid formed. Confirm your answer by clicking the 'checkbox'.
- A mould for producing drinking glasses is to be made by rotating a section of the curve \(y = {x^2}\) about the \(y\)-axis. The design specifications say that each glass must have a base diameter of 4 cm and a capacity not less than 282 mL. To meet the design specifications, what would the minimum height of the glass need to be?
- What would be the capacity of a glass with the same height and base diameter as the glass in question (3), but formed by rotating the curve \(y = {x^3}\) about the \(y\)-axis? [Hint: To find the lower limit of the integral, sketch the curve \(y = {x^3}\) and locate the value of \(y\), such that the diameter of the base of the glass is 4 cm.]