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Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = x³ y = 16x about the x-axis.

User Swimburger
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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.

User Vladimir Muzhilov
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