Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1.
ANSWER:
v = ∫[1,e] π(4-(lnx+1)^2) dx
looks good to me. Or, using shells,
v = ∫[0,1] 2π(y+1)(e^y-1) dy
thank you!
To set up the integral, follow these steps:
1. Identify the limits of integration: The region is bounded by the curves y = Ln(x), y = 1, and x = 1. In this case, the lower limit of integration is x = 1 since the region starts at x = 1. The upper limit of integration is x = e, where e is the natural logarithm base, since the region ends where y = Ln(x) intersects with y = 1.
2. Determine the radius of each cross-sectional disk: In this problem, the line of revolution is y = -1. To find the radius of each disk, subtract the distance between the line y = -1 and the curve y = Ln(x) from the distance between the line y = -1 and the curve y = 1. Since the curve y = Ln(x) is the closest to the line of revolution, the distance is Ln(x) + 1.
3. Find the area of each cross-sectional disk: The area of each disk is π times the square of the radius. In this case, the radius is (4 - (Ln(x) + 1)^2), which gives us the area of each disk as π(4 - (Ln(x) + 1)^2).
4. Set up the integral: To find the volume, integrate the area of each disk over the range of x values from 1 to e. The integral is given by:
v = ∫[1,e] π(4 - (Ln(x) + 1)^2) dx
Note that you would evaluate this integral numerically to find the actual volume.
To set up the integral, we need to find the cross-sectional area of the solid generated when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1.
Step 1: Determine the range of integration:
The region bounded by the curves y = Ln(x), y = 1, and x = 1 is confined between x = 1 and x = e (where e is the base of the natural logarithm).
Step 2: Determine the radius of each cross-section:
The cross-sections of the solid are disks with radii defined by the distance between the curve y = Ln(x) and the line y = -1. At any given x-value, the radius is the difference between the y-value of Ln(x) and -1.
Radius = Ln(x) - (-1) = Ln(x) + 1
Step 3: Determine the area of each cross-section:
The area of each cross-section is given by the formula for the area of a circle, which is πr^2, where r is the radius. In this case, the radius is Ln(x) + 1.
Area = π(Ln(x) + 1)^2
Step 4: Set up the integral:
Since we are finding the volume, we need to integrate the areas of the cross-sections over the range of integration.
v = ∫[1,e] π(Ln(x) + 1)^2 dx
This integral represents the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1. However, we don't evaluate this integral.