Question

Suppose the base of a solid S is an elliptical region with boundary 25 x^2 +9 y^2 = 225 and its cross-sections perpendicular to the x-axis are squares. Find the volume of S. Give an exact expression for your answer.

Answer 1

To find the volume of the solid S, we integrate the area of the squares formed by cross-sections perpendicular to the x-axis over the range of x-values that the ellipse covers.

Step-by-step explanation:

To find the volume of the solid S, we need to consider the cross-sections perpendicular to the x-axis. Since the cross-sections are squares, we can find the area of one of these squares and then integrate it over the range of x-values that the ellipse covers.

The equation of the ellipse is 25x^2 + 9y^2 = 225. Solving for y, we get y = sqrt(225 - (25x^2/9)).

The area of the square is equal to the side length squared, which is 2y. So, the volume of the solid S can be found by integrating 4y^2 over the range of x-values that the ellipse covers.