Standard +0.3 This is a straightforward volume of revolution problem requiring students to set up and evaluate ∫π y² dx from x=1 to x=2, where y²=16-x². The integration is elementary (polynomial), and the setup follows a standard template. Slightly above average difficulty only due to the bounds being away from the origin, but still a routine calculus exercise.
A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]
Attempt use of $(\pi) \int [16 - x^2] dx$ | B1 |
Attempt integration | M1 | At least one power must rise in their single variable integral
Obtain $(\pi) \left[ 16x - \frac{1}{3}x^3 \right]$ | A1 |
Use of correct limits in correct order | M1 |
Obtain $\frac{41\pi}{3}$ or 42.9 or better | A1 | [5] |
A circle has equation $x^2 + y^2 = 16$. Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines $x = 1$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. [5]
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q5 [5]}}