Find the volume of the solid formed by rotating the region enclosed by y=e^(3x)+2, y=0, x=0, and x=.2 about the y-axis
To find the volume of a solid formed by rotating a region about the y-axis, we can use the method of cylindrical shells. Here's how you can do it step by step:
Step 1: Visualize the region
Plot the two given curves, y = e^(3x) + 2 and y = 0, on a coordinate plane. The region enclosed by these curves, x = 0 to x = 0.2, should form a bounded shape. Make sure you have a clear picture of the region in your mind.
Step 2: Set up the integral
The volume of the solid can be found by integrating the circumference of each cylindrical shell multiplied by its height. Each cylindrical shell is formed by taking an infinitesimally thin strip of width dx along the x-axis and rotating it about the y-axis.
The height of each cylindrical shell is the difference between the two curves, which is given by y = e^(3x) + 2 - 0 = e^(3x) + 2.
The circumference of each cylindrical shell is given by 2πy, where y is the distance from the y-axis to the curve.
Therefore, the integral to find the volume is: V = ∫[a,b] 2πy * dx, where [a,b] represents the interval of x-values.
Step 3: Determine the limits of integration
In this case, the region is bounded by x = 0 and x = 0.2, so the limits of integration are x = 0 to x = 0.2.
Step 4: Evaluate the integral
Plug in the expression for y: y = e^(3x) + 2.
The integral to find the volume becomes:
V = ∫[0,0.2] 2π(e^(3x) + 2) dx.
Step 5: Compute the integral
Integrate the function 2π(e^(3x) + 2) with respect to x over the limits [0,0.2]:
V = 2π ∫[0,0.2] (e^(3x) + 2) dx.
To solve this integral, integrate e^(3x) first and then integrate 2 separately over the same limits.
∫e^(3x) dx = (1/3)e^(3x) (integration of e^u is e^u/du, where u = 3x and du = 3dx).
∫2 dx = 2x.
Substituting the limits, we have:
V = 2π [(1/3)e^(3x)] + 2x | [0,0.2]
V = 2π [(1/3)e^(0.6) + 2(0.2)] - 2π(0 + 0)
V = 2π (e^(0.6)/3 + 0.4)
Step 6: Calculate the volume
Approximate the value of e^(0.6) using a calculator (it's approximately 1.822).
V ≈ 2π (1.822/3 + 0.4)
V ≈ 2π (0.607 + 0.4)
V ≈ 2π (1.007)
V ≈ 6.334π
So, the volume of the solid formed by rotating the given region about the y-axis is approximately 6.334π cubic units.