Final answer:
To find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = x³ and y = 16x about the x-axis, you can use the method of cylindrical shells. The volume is approximately 1635.14 cubic units.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region inside the first quadrant enclosed by the curves y = x³ and y = 16x about the x-axis, we can use the method of cylindrical shells.
First, let's determine the intersection points of the two curves:
- Solving y = x³ and y = 16x, we get x³ = 16x.
- Dividing both sides by x gives x² = 16.
- Taking the square root of both sides gives x = ±4.
Since we are only interested in the region inside the first quadrant, we take the positive value of x, which is x = 4.
To find the volume, we integrate the circumference of each shell multiplied by its height. The height of each shell is given by the difference in the y-values of the two curves at a particular x-value. The radius of each shell is the x-value itself.
The integral setup is as follows:
V = ∫[0, 4] 2πx (16x - x³) dx
Simplifying the expression and evaluating the integral, we find that the volume of the solid is approximately 1635.14 cubic units.