Final answer:
To find the volumes of the solids bounded by the circle x^2 + y^2 = 25, with cross sections taken perpendicular to the x-axis, we can think of each cross section as a disk. Using the equation of the circle, we can determine the radius of each disk and find the volume of each disk using the formula for the volume of a cylinder. The total volume is found by integrating the volumes of the disks over the range of the circle.
Step-by-step explanation:
To find the volume of the solids whose bases are bounded by the circle x^2 + y^2 = 25, with cross sections taken perpendicular to the x-axis, we can think of each cross section as a disk. The radius of each disk will be determined by the equation of the circle, which is r = sqrt(25 - x^2). The height of each disk will be dx, as we can consider infinitesimally small widths along the x-axis. Therefore, the volume of each disk is given by dV = π * (sqrt(25 - x^2))^2 * dx.
To find the total volume, we need to integrate the volumes of the disks from x = -5 to x = 5, which corresponds to the entire range of the circle. So, the total volume is given by V = ∫[from -5 to 5] π * (sqrt(25 - x^2))^2 * dx.
Simplifying the integral, we can write it as V = ∫[from -5 to 5] π * (25 - x^2) * dx. Evaluating this integral will give us the volume of the solids bounded by the given circle.