Answer:
Explanation:
The volume of a solid obtained by rotating a two-dimensional region about an axis can be found using the method of cylindrical shells.
The region in question is bounded by the parabolic curve y = x^2, the vertical line x = 5, and the x-axis. We can find the volume of the solid obtained by rotating this region about the x-axis by adding up the volumes of an infinite number of infinitely thin cylindrical shells with radius equal to the distance from the axis to the boundary of the region at each y-value.
The formula for the volume of a cylindrical shell is given by V = 2 * pi * y * dV, where y is the distance from the x-axis to the boundary of the region at each y-value, and dV is the incremental volume at that y-value.
To find the volume, we need to find the incremental volume dV, and integrate the cylindrical shell formula over the region bounded by y = x^2 and x = 5. The incremental volume is given by dV = pi * (R^2 - r^2)dy, where R is the outer radius, r is the inner radius, and dy is the incremental change in y.
In this case, the outer radius R is equal to the value of x = 5 when y = x^2, and the inner radius is equal to zero. Thus, the incremental volume is given by: dV = pi * (5^2 - 0^2)dy = 25 * pi * dy.
The bounds of the integration are from 0 to 5^2 = 25. Integrating the cylindrical shell formula over this range, we find that the volume of the solid obtained by rotating the region bounded by y = x^2, x = 5, and y = 0 about the x-axis is:
V = 2 * pi * integral from 0 to 25 of (y * 25 * pi)dy = 2 * pi * (25^2 * 25 / 2) = 15625 * pi cubic units.
So, the volume of the solid is approximately 49,093 cubic units.