find the volume of the solid formed by revolving the region bounded by y=e^x, y=0, x=o and x=1 about the y axis

by using the shell method, i got

v=2*pi int(o to 1) (x)(e^x) dx

and my teacher said the answer is e/2 but i keep getting a different answer

Are you sure it is to be rotated about the y axis?
Your attempt ends in dx suggesting you are rotating about the x axis.

It would be a relatively easy question if it is rotated about the x axis, but if you rotate about the y axis that would involve the integral of Pi*(ln y)^2 dy from 1 to e

let me know which way it is before I do that difficult question.

Based on the problem statement, we are revolving the region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 about the y-axis. To find the volume using the shell method, we integrate the product of the circumference of a shell (2πrh) and the height of the shell (dx).

The height of the shell, dx, represents an infinitesimally small change in x as we move from x to x + dx. The radius of the shell, r, is equal to x since we are revolving the region about the y-axis. The circumference of the shell is 2πrh = 2πx(e^x).

Thus, the formula for the volume using the shell method becomes:

V = ∫[from x = 0 to x = 1] 2πx(e^x) dx

Evaluating this integral will give us the volume of the solid formed by revolving the region about the y-axis.

However, it seems you made a mistake in your expression. Instead of v = 2π ∫[from 0 to 1] (x)(e^x) dx, the correct integral should be:

V = 2π ∫[from 0 to 1] (x)(e^x) dx

Now, let's evaluate this integral to find the answer:

V = 2π ∫[from 0 to 1] (x)(e^x) dx

To evaluate this integral, you can use integration by parts. Taking u = x and dv = e^x dx, we have du = dx and v = e^x. Applying integration by parts, we get:

V = 2π [(x)(e^x) - ∫[from 0 to 1] e^x dx]

Now integrate the remaining term:

V = 2π [(x)(e^x) - e^x] [from 0 to 1]
= 2π [(1)(e^1) - e^1] - 2π [(0)(e^0) - e^0]
= 2π [e - e] - 2π [0 - 1]
= 2π (0) - 2π (-1)
= 2π

Hence, the volume of the solid formed by revolving the region bounded by y = e^x, y = 0, x = 0, and x = 1 about the y-axis is 2π.

It seems that the answer provided by your teacher as e/2 is incorrect.