- #1
Theia
- 122
- 1
Let \(\displaystyle f(x) = x^3 + 4x^2 - x + 5\) revolve about the line \(\displaystyle y(x) = -x + 5\). There will form one solid with finite volume. Find the volume of that solid.
The formula for finding the volume of a solid of revolution is V = ∫(πf(x))^2 dx, where f(x) is the function being rotated around the axis of revolution.
A solid of revolution is created by rotating a 2-dimensional shape around an axis, while a solid of extrusion is created by extending a 2-dimensional shape in the third dimension. In other words, a solid of revolution is a 3-dimensional object, while a solid of extrusion is still a 2-dimensional object that has been extended.
The choice of axis of revolution does not affect the volume of the solid of revolution. As long as the same function is being rotated around the axis, the volume will remain the same.
The shape of the 2-dimensional cross section will determine the shape of the resulting solid of revolution. For example, if the cross section is a circle, the resulting solid will be a sphere. If the cross section is a square, the resulting solid will be a cube.
No, the volume of a solid of revolution cannot be negative. It is always a positive value, representing the amount of space occupied by the solid object.