The diagram shows the curve $$\(y=2+\mathrm{e}^{-2 x}\)$$. The curve crosses the $$\(y\)$$-axis at the point $$\(A\)$$, and the point $$\(B\)$$ on the curve has $$\(x\)$$-coordinate 1 . The shaded region is bounded by the curve and the line segment $$\(A B\)$$. Find the exact area of the shaded region. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_w20_qp_21 Year：2020 Question No：3
Answer:
Integrate to obtain form \(a x+b \mathrm{e}^{-2 x}\)
Obtain correct \(2 x-\frac{1}{2} \mathrm{e}^{-2 x}\)
Apply limits to obtain \(\frac{5}{2}-\frac{1}{2} \mathrm{e}^{-2}\)
Attempt to find area of relevant trapezium
Obtain \(\frac{5}{2}+\frac{1}{2} \mathrm{e}^{-2}\) and subtract to obtain \(\mathrm{e}^{-2}\) or exact equivalent
Obtain correct \(2 x-\frac{1}{2} \mathrm{e}^{-2 x}\)
Apply limits to obtain \(\frac{5}{2}-\frac{1}{2} \mathrm{e}^{-2}\)
Attempt to find area of relevant trapezium
Obtain \(\frac{5}{2}+\frac{1}{2} \mathrm{e}^{-2}\) and subtract to obtain \(\mathrm{e}^{-2}\) or exact equivalent
Knowledge points:
2.5.1 extend the idea of 'reverse differentiation' to include the integration of (Knowledge of the general method of integration by substitution is not required.)
Solution:
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