1.6: Intersection and enclosed area of exponential curves

4PM1/2R/June/2022 — Question 8 · 11 marks

1.6 diagram 1
Figure 3 shows a sketch of part of the curves with equations
y=e3x−1andy=9−9e−3x.y=\mathrm{e}^{3x}-1 \qquad \text{and} \qquad y=9-9\mathrm{e}^{-3x}.
The curves intersect at the points A\displaystyle A and B\displaystyle B, as shown in Figure 3.
(a) (i) Show that the x\displaystyle x coordinates of A\displaystyle A and B\displaystyle B satisfy the equation
(e3x)2−10e3x+9=0.(\mathrm{e}^{3x})^2-10\mathrm{e}^{3x}+9=0.
(2)
(ii) Hence show that the x\displaystyle x coordinate of B\displaystyle B is 13ln 9\displaystyle \frac{1}{3}\mathrm{ln}\,9.
(3)
(b) Find the exact area of the finite region bounded by the two curves.
(6)