Consider the region bounded by the curves y=e^x, y=-e^x, and x=1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating this region about the y-axis.
To find the volume using the cylindrical shells method, we need to integrate the shell volume over the given region.
The region is bounded by the curves y=e^x, y=-e^x, and x=1. Let's analyze this region before setting up the integral.
First, we can see that the x-value ranges from 0 to 1, as given by the curve x=1. This will be the range of our integral.
Now, let's consider a vertical strip at an arbitrary x-value between 0 and 1. The height of this strip will be the difference between the function values y=e^x and y=-e^x at that x-coordinate, which is 2e^x.
To calculate the radius of each cylindrical shell, we take the x-coordinate as the radius since we're rotating the region about the y-axis.
The volume of each cylindrical shell is given by 2πrh, where r is the radius and h is the height.
Therefore, the volume of the solid obtained by rotating this region about the y-axis can be found by integrating the volume of each cylindrical shell over the range of x-values.
Let's set up the integral:
V = ∫(from x=0 to x=1) 2π(x)(2e^x) dx
To find the integral, we can simplify it:
V = 4π∫(from x=0 to x=1) xe^x dx
To solve this integral, we use integration by parts. Let's assign u = x and dv = e^x dx.
Differentiating u, we get du = dx, and integrating dv, we get v = e^x.
Now, we can use the integration by parts formula:
∫uv dx = uv - ∫vu' dx
Applying this formula to our integral:
V = 4π[xe^x - ∫e^x dx] (from x=0 to x=1)
Evaluating the integral and plugging in the limits:
V = 4π[(1)(e^1 - 1) - (0)(e^0 - 1)]
Simplifying, we get:
V = 4π(e - 1)
Therefore, the volume of the solid obtained by rotating the given region about the y-axis is 4π(e - 1) cubic units.