JEE Advanced · Mathematics · 28. Area Under Curves
Area of the region bounded by the curve \(y=e^x\) and lines \(x=0\) and \(y=e\) is
- A
\(e-1\)
- B
\(\int_1^e \ln (e+1-y) d y\)
- C
\(e-\int_0^1 e^x d x\)
- D
\(\int_1^e \ln y d y\)
Answer & Solution
Correct Answer
(B)
\(\int_1^e \ln (e+1-y) d y\)
Step-by-step Solution
Detailed explanation
Shaded area \(=e-\left(\int_0^1 e^x d x\right)=1\)
Also, \(\quad \int_1^e \ln (e+1-y) d y\)
\[
\begin{aligned}
& =\int_e^q \ln t(-d t) \\
& \text { [put } e+1-y=t \Rightarrow-d y=d t \text { ] } \\
& =\int_1^e \ln t d t=\int_1^e \ln y d y=1
\end{aligned}
\]
Also, \(\quad \int_1^e \ln (e+1-y) d y\)
\[
\begin{aligned}
& =\int_e^q \ln t(-d t) \\
& \text { [put } e+1-y=t \Rightarrow-d y=d t \text { ] } \\
& =\int_1^e \ln t d t=\int_1^e \ln y d y=1
\end{aligned}
\]
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