Standard +0.3 This is a straightforward volume of revolution question requiring integration of y² between given limits. The main steps are: set up the integral π∫[1 to 2.4](1/(5x-4)³)² dx, use substitution u=5x-4, and evaluate. While it requires careful algebraic manipulation and substitution technique, it follows a standard template with no conceptual surprises, making it slightly easier than average.
\includegraphics{figure_9}
The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\).
Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]
May be done using 1. This would be the only mark
1
available if candidate integrates y.
1
Volume under curve =π dx
5x42
Answer
Marks
Guidance
3
M1
No further marks available if y.
3 1
π 5x4
= 3
Answer
Marks
Guidance
5
B1 B1
Calculator used for integration scores no further marks.
3 1 3
=π 83 1 π
Answer
Marks
Guidance
5 5
M1
Uses limits 1, 2.4 in an integral of y2.
7 3 4
Volume = π π = π
Answer
Marks
Guidance
5 5 5
A1
SC B1 if the only error is not showing substitution.
6
Answer
Marks
Guidance
Question
Answer
Marks
Question 9:
9 | 7 7
Volume of cylinder = π12 π
5 5 | B1 | 2.4
May be done using 1. This would be the only mark
1
available if candidate integrates y.
1
Volume under curve =π dx
5x42
3 | M1 | No further marks available if y.
3 1
π 5x4
= 3
5 | B1 B1 | Calculator used for integration scores no further marks.
3 1 3
=π 83 1 π
5 5 | M1 | Uses limits 1, 2.4 in an integral of y2.
7 3 4
Volume = π π = π
5 5 5 | A1 | SC B1 if the only error is not showing substitution.
6
Question | Answer | Marks | Guidance
\includegraphics{figure_9}
The diagram shows part of the curve with equation $y = \frac{1}{(5x - 4)^3}$ and the lines $x = 2.4$ and $y = 1$. The curve intersects the line $y = 1$ at the point $(1, 1)$.
Find the exact volume of the solid generated when the shaded region is rotated through $360°$ about the $x$-axis. [6]
\hfill \mbox{\textit{CAIE P1 2024 Q9 [6]}}