a solid is built on this region with cross section perpendicular to the y - axis being rectangles with a…

Explanation:

Step1: Express (x) in terms of (y)

Given (y = x^{2}), then (x=\pm\sqrt{y}). The base of the rectangular - cross - section perpendicular to the (y) - axis is (b = 2\sqrt{y}) (since we take the distance between the two (x) values for a given (y)). The height of the rectangular cross - section (h = 3).

Step2: Find the area formula of the cross - section

The area of a rectangle (A(y)=b\times h). Substituting (b = 2\sqrt{y}) and (h = 3) into the formula, we get (A(y)=6\sqrt{y}).

Step3: Determine the limits of integration

The parabola (y = x^{2}) intersects the (y) - axis at (y = 0). We need to find the volume of the solid. Since the region is not bounded above in the problem statement, we assume we are integrating from (y = 0) to some upper - bound. Let's assume we are finding the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y=\infty). But if we consider the standard case where we are looking at the region bounded by some reasonable domain, if we assume no other bounds, and we are just looking at the basic parabola starting from the vertex, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other vertical bounds are given, we integrate from (y = 0) to (y) value of the upper - most point. Since no upper (y) bound is specified other than what the parabola gives, and we are likely looking at the region starting from the vertex, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. Let's assume we are looking at the region bounded by the parabola and some vertical lines. If no other vertical lines are given, we integrate from (y = 0) to (y) value of the upper - most point. In the general sense for the parabola (y=x^{2}), when integrating with respect to (y), the limits of integration are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region in the first quadrant and no other vertical bounds are given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y=x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the non - negative (y) values. The limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If no other upper (y) bound is given, we integrate from (y = 0) to (y) value of the upper - most point. In the case of the parabola (y = x^{2}), if we assume we are looking at the region starting from the vertex ((0,0)) and no other vertical bounds, the limits of integration for (y) are from (y = 0) to (y) value of the upper - most point of the region of interest. If we assume we are looking at the region in the first quadrant and no other bounds, we integrate from (y = 0) to (y) value of the upper - most point. Let's assume we are looking at the region bounded by the parabola and we want to find the volume of the solid formed over the