Question

Consider the region bounded by the graphs of x = y^2 and x = 9. Find the volume of the solid that has this region as its base if every cross section by a plane perpendicular to the x-axis has the shape of an equilateral triangle.

Answer 1

To find the volume of the solid, integrate the area of each equilateral triangle cross section over the range of x = 0 to x = 9.

Step-by-step explanation:

The region bounded by the graphs of x = y^2 and x = 9 forms a parabolic shape between x = 0 and x = 9, where the parabola opens to the right.

To find the volume of the solid that has this region as its base, we need to consider the cross sections formed by planes perpendicular to the x-axis, which have the shape of equilateral triangles.

For any given x-value, the height of the equilateral triangle cross section is equal to the y-value at that x-value, since the sides of the triangle are defined by the equation x = y^2.

The base of the equilateral triangle is equal to the width of the region, which is 9 units.

Therefore, the volume of the solid can be found by integrating the area of each equilateral triangle cross section over the range of x = 0 to x = 9:

V = ∫(0 to 9) (1/2) * (9) * (y^2) * √3 * (y^2) dy