Standard +0.8 This is a solid of revolution problem requiring integration of multiple sections (line segments and a circular arc) rotated about the y-axis. Students must set up and evaluate ∫πx²dy for three distinct regions, including finding x² from the circle equation x² = 37 - (y-10)². While the individual integrations are standard, coordinating multiple pieces and managing the algebra elevates this above routine questions.
\(O\) is the origin of a coordinate system whose units are cm. The points \(A\), \(B\), \(C\) and \(D\) have coordinates \((1, 0)\), \((1, 4)\), \((6, 9)\) and \((0, 9)\) respectively.
The arc \(BC\) is part of the curve with equation \(x^2 + (y - 10)^2 = 37\).
The closed shape \(OABCD\) is formed, in turn, from the line segments \(OA\) and \(AB\), the arc \(BC\) and the line segments \(CD\) and \(DO\) (see diagram).
A funnel can be modelled by rotating \(OABCD\) by \(2\pi\) radians about the \(y\)-axis.
\includegraphics{figure_6}
Find the volume of the funnel according to the model. [3]
Question 6:
6 | For AB,V =π×12×4=12.566... ( )
For BC, V =∫ b πx2dy=π∫ 9( 37−( y−10 )2 ) dy
a 4
=356.05....
⇒Total V =356.05...+12.566...=368.61...
( )
=369 cm3 to3sf | M1
A1
A1 | 3.3
1.1
3.4 | 4π
Split into two parts and use formulae
An integral and an attempt at the volume of a
cylinder must be seen
Integration – ignore limits BC
340π
3
Units are not required
352π
3
[3]
$O$ is the origin of a coordinate system whose units are cm. The points $A$, $B$, $C$ and $D$ have coordinates $(1, 0)$, $(1, 4)$, $(6, 9)$ and $(0, 9)$ respectively.
The arc $BC$ is part of the curve with equation $x^2 + (y - 10)^2 = 37$.
The closed shape $OABCD$ is formed, in turn, from the line segments $OA$ and $AB$, the arc $BC$ and the line segments $CD$ and $DO$ (see diagram).
A funnel can be modelled by rotating $OABCD$ by $2\pi$ radians about the $y$-axis.
\includegraphics{figure_6}
Find the volume of the funnel according to the model. [3]
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q6 [3]}}