Final answer:
The volume of the solid obtained by rotating the region between y = 5 sin(x) and y = 5 cos(x) around the line y = -1 can be calculated using integral calculus, taking into account the symmetry of sin² and cos² over a complete cycle.
Step-by-step explanation:
The volume V of a solid created by rotating a region around a line can be found using the disk or washer method from calculus. In this case, we are rotating the region bounded by y = 5 sin(x) and y = 5 cos(x) around the line y = -1.
The region is rotated about this line from x = 0 to x = π/2, where both functions complete a quarter cycle and intersect. From the formula:
V = π ∫ (R² - r²) dx
where
R and r are the outer and inner radii from the line of revolution to the curves
the volume can be calculated.
However, since the sine and cosine functions are merely phase-shifted versions of each other, their squares will result in the same average value over a complete cycle.
Therefore, the integral will involve cos²(x) or sin²(x) and their average value over the interval, which will simplify the calculation of the volume.