Question

The area bounded by the curves y = log_ex and y = (log_ex)² is ______  

Options

• 3 - e

• e - 3

• `1/2(3 - e)`

• `1/2(e - 3)`

Answer 1

The area bounded by the curves y = log_ex and y = (log_ex)² is 3 - e.

Explanation:

 

Required area = `int_1^e [logx - (logx)^2]dx`

= `int_1^e log x dx - int_1^e (logx)^2 dx`

= `[xlogx - x]_1^e - [x(logx)^2 - 2xlogx + 2x]_1^e`

= [e - e - (-1)] - [e(1)² - 2e + 2e - (2)]

= 1 - (e - 2)

= 3 - e