Oxford University MAT 2021 Solved Papers

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Question : 4 of 10

Marks: +1, -0

The area of the region bounded by the curve

y=e^x

, the curve

y=1-e^x

, and the

y

-axis equals

0 ,
$1-\ln 2$ ,
$\;{1}/{2}-\;{1}/{2} \ln 2$ ,
$\ln 2-1$ ,
$1-\ln \;{1}/{2}$ .

Solution:

The intersection point is at

e^x=1-e^x

which happens when

2 e^x=1

, that is

x=-\ln 2< 0

.

The area has reflectional symmetry in the line

y=\;{1}/{2}

so we want

2 ∫↙{-\ln 2}^0 e^x-\;{1}/{2} {d} x=2[e^x-\;{x}/{2}]_{-\ln 2}^0=2((1)-(\;{1}/{2}+\;{\ln 2}/{2}))=1-\ln 2

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