AQA
Further Paper 1
2023
June
— Question 16
7 marks
Exam Board
AQA
Module
Further Paper 1 (Further Paper 1)
Year
2023
Session
June
Marks
7
Topic
Second order differential equations
16
Show that
$$\int _ { 0.5 } ^ { 4 } \frac { 1 } { t } \ln t \mathrm {~d} t = a ( \ln 2 ) ^ { 2 }$$
where \(a\) is a rational number to be found.
16
A curve \(C\) is defined parametrically for \(t > 0\) by
$$x = 2 t \quad y = \frac { 1 } { 2 } t ^ { 2 } - \ln t$$
The arc formed by the graph of \(C\) from \(t = 0.5\) to \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to generate a surface with area \(S\)
Find the exact value of \(S\), giving your answer in the form
$$S = \pi \left( b + c \ln 2 + d ( \ln 2 ) ^ { 2 } \right)$$
where \(b , c\) and \(d\) are rational numbers to be found. [0pt]
[7 marks]