find area bounded by the curve originate and x-axis y = lnx, x = 1 , x = e​

To find the area bounded by the curve y = ln(x), the x-axis, and the lines x = 1 and x = e, we can use the concept of definite integrals.

The area A under the curve y = f(x) from x = a to x = b is given by the definite integral:

A = ∫ from a to b f(x) dx

Here, our f(x) is ln(x), a is 1, and b is e.

So, we need to calculate the integral:

A = ∫ from 1 to e ln(x) dx

To solve this integral, we can use integration by parts, where u = ln(x) and dv = dx.

Then, du = (1/x) dx and v = x.

According to the formula for integration by parts, ∫ u dv = u*v - ∫ v du.

So, we get:

A = ln(x)x - ∫ x(1/x) dx from 1 to e = ln(x)x - ∫ dx from 1 to e = ln(x)x - x from 1 to e = [eln(e) - e] - [1ln(1) - 1] = [e*1 - e] - [0 - 1] = e - e + 1 = 1

So, the area bounded by the curve y = ln(x), the x-axis, and the lines x = 1 and x = e is 1 square unit.