Find the volume of the solid generated by rotating the region bounded by the graph of y=x^2, y=0, x=1, and x=3 about the line parallel to the x-axis and one unit below it.
Please help I have no idea how to solve this problem.
To find the volume of the solid generated by rotating a region about a line, we can use the method of cylindrical shells.
Here's how you can solve the problem step by step:
1. First, sketch the given region. In this case, the region is bounded by the graph of y = x^2, y = 0, x = 1, and x = 3.
2. Determine the axis of rotation. In this case, the line parallel to the x-axis and one unit below it will serve as the axis of rotation.
3. Visualize the solid formed by rotating the region. In this case, we will rotate the region about the axis of rotation to form a three-dimensional shape with a hollow center.
4. Set up the integral for the volume using cylindrical shells. The general formula for the volume of a solid obtained by rotating a region about a line is given by:
V = ∫(2πrh) dx
where r is the distance from the axis of rotation to the shell, h is the height of the shell, and dx is the width of the shell.
5. Calculate the radius and height of the shell at a generic x-coordinate within the region. Since we are rotating about the x-axis, the radius of the shell is x units, and the height is y units. In this case, y = x^2.
6. Substitute the values of r, h, and dx into the volume formula. The volume element becomes:
dV = 2π(x)(x^2) dx
7. Determine the limits of integration. In this case, the region is bounded by x = 1 and x = 3, so the integral will be evaluated over this interval.
8. Evaluate the integral. Integrate the expression 2πx^3 with respect to x from 1 to 3:
V = ∫[1 to 3] (2πx^3) dx
9. Calculate the definite integral. Evaluating the integral gives the volume of the solid:
V = [2πx^4/4] from 1 to 3
= (π/2)(3^4 - 1^4)
= (π/2)(81 - 1)
= (π/2)(80)
= 40π
10. Simplify the expression. The final volume is 40π cubic units.
Therefore, the volume of the solid generated by rotating the given region about the line parallel to the x-axis and one unit below it is 40π cubic units.