The region enclosed by the curve y =e^x, the x-axis, and the lines x=0 and x=1 is revolved about the x-axis. Find the volume of the resulting solid formed.

just use discs:

v = integral(pi*r^2)dx from x=0 to 1
where r = y = e^x
v = pi*Integral(e^2x dx) [0,1]
= pi(1/2 e^2x) [0,1]
= pi(e^2/2 - 1/2)
= pi/2 (e^2 - 1)

To find the volume of the solid formed by revolving the region enclosed by the curve y = e^x, the x-axis, and the lines x = 0 and x = 1 about the x-axis, we can use the method of cylindrical shells.

Step 1: Set up the integral to calculate the volume using cylindrical shells. The general formula for the volume of a solid obtained by revolving a curve y = f(x) around the x-axis from x = a to x = b is:

V = 2π ∫[a, b] x * f(x) dx

In this case, the bounds of integration are from x = 0 to x = 1, and f(x) = e^x. Therefore, we have:

V = 2π ∫[0, 1] x * e^x dx

Step 2: Calculate the integral. To evaluate this integral, we can use integration by parts. The integration by parts formula states:

∫ u dv = uv - ∫v du

Letting u = x and dv = e^x dx, we can calculate du and v:

du = dx
v = ∫e^x dx = e^x

Applying the integration by parts formula:

∫ x * e^x dx = x * e^x - ∫e^x dx

Simplifying:

∫ x * e^x dx = x * e^x - e^x + C

Now we can evaluate the integral:

V = 2π [ (x * e^x - e^x) ∣[0, 1] ]
= 2π [(1 * e^1 - e^1) - (0 * e^0 - e^0)]
= 2π [e - e^1]
= 2π (e - e)
= 0

Therefore, the volume of the resulting solid formed is 0.