1.4: Volume of revolution bounded by exponential curves

4PM1/2/June/2021 — Question 11 · 8 marks

1.4 diagram 1
The region R\displaystyle R, shown shaded in Figure 4, is bounded by the curve with equation
y=ex,y=\mathrm{e}^x,
the curve with equation
y=4e−x,y=4\mathrm{e}^{-x},
the straight line with equation x=a\displaystyle x=a, the x\displaystyle x-axis and the y\displaystyle y-axis.
When the region R\displaystyle R is rotated through 360∘\displaystyle 360^\circ about the x\displaystyle x-axis, the volume of the solid generated is
k−8πe−4,k-8\pi\mathrm{e}^{-4},
where k\displaystyle k is a constant.
Using algebraic integration, find a possible value of a\displaystyle a and the exact corresponding value of k\displaystyle k.
(8)