1.23: Figure 4 shows part of a curve with equation .

4PM1/2R/June/2025 — Question 11 · 12 marks

1.23 diagram 1
Figure 4 shows part of a curve C\displaystyle C with equation y=3ex3\displaystyle y = 3\mathrm{e}^{\frac{x}{3}}.
The point A\displaystyle A with coordinates (2,p)\displaystyle (2, p) lies on C\displaystyle C
(a) Write down the exact value of p\displaystyle p
(1)
The straight line n\displaystyle n, shown on Figure 4, is the normal to C\displaystyle C at A\displaystyle A and crosses the x\displaystyle x-axis at point B\displaystyle B.
The region R\displaystyle R, shown shaded in Figure 4, is bounded by C\displaystyle C, n\displaystyle n, the x\displaystyle x-axis and the y\displaystyle y-axis.
(b) Use algebraic integration to find the area of R\displaystyle R
Give your answer in the form WVe2+We23−W\displaystyle \frac{W}{V} \mathrm{e}^2 + W \mathrm{e}^{\frac{2}{3}} - W
where W\displaystyle W and V\displaystyle V are integers to be found.
(11)