Standard +0.3 This is a straightforward volumes of revolution question requiring students to set up and evaluate a standard integral about the y-axis. The function inverts easily (x² = y - 1), and the integration ∫(y - 1)dy is routine. While it requires correct formula application and careful bounds handling, it involves no conceptual challenges beyond standard technique, making it slightly easier than average.
\includegraphics{figure_1}
The diagram shows part of the curve \(y = x^2 + 1\). Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [4]
Integral is \(\frac{1}{3}y^3 - y\) or \(\frac{(y-1)^3}{2}\)
Limits for \(y\) are 1 to 5
Answer
Marks
Guidance
\(\rightarrow 8\pi\) or \(25.1\) (AWT)
M1, A1, B1, A1 [4]
Use of \([x^3 - \text{not}]y^2\) – ignore \(\pi\) co; Sight of an integral sign with 1 and 5; co (no \(\pi\) max 3/4)
$\text{Vol} = (\pi) \int_1^5 y^2 \, dy = (\pi) \int (y-1) \, dy$
Integral is $\frac{1}{3}y^3 - y$ or $\frac{(y-1)^3}{2}$
Limits for $y$ are 1 to 5
$\rightarrow 8\pi$ or $25.1$ (AWT) | M1, A1, B1, A1 [4] | Use of $[x^3 - \text{not}]y^2$ – ignore $\pi$ co; Sight of an integral sign with 1 and 5; co (no $\pi$ max 3/4)
\includegraphics{figure_1}
The diagram shows part of the curve $y = x^2 + 1$. Find the volume obtained when the shaded region is rotated through $360°$ about the $y$-axis. [4]
\hfill \mbox{\textit{CAIE P1 2014 Q1 [4]}}