The base of a solid is a region located in quadrant 1 that is bounded by the axes, the graph of y = x^2 - 1, and the line x = 2. If cross-sections perpendicular to the x-axis are squares, what would be the volume of this solid?
Hmmm. The y-axis does not form part of the boundary.
Each square of thickness dx has a base that is y=x^2-1 in width.
So, adding up all the squares, the volume v is
∫[1,2] (x^2-1)^2 dx = 38/15
To find the volume of the solid, we need to integrate the areas of the cross-sections parallel to the x-axis over the interval where the solid exists.
First, let's find the limits of integration for the x-axis. The solid is bounded by the axes, the graph of y = x^2 - 1, and the line x = 2. To find the limits, we need to find the intersection points of these curves.
The curve y = x^2 - 1 intersects the x-axis when y = 0:
0 = x^2 - 1
x^2 = 1
x = ±1
So, the limits of integration for the x-axis are x = -1 and x = 2.
Next, let's consider a cross-section perpendicular to the x-axis at a specific value of x. Since the cross-sections are squares, the area of each cross-section is equal to the side length squared.
The side length of the cross-section at any x-coordinate is the difference between the y-coordinate values of the upper and lower boundaries of the solid.
The upper boundary is given by the graph of y = x^2 - 1, and the lower boundary is the x-axis.
So, the side length of the cross-section at x is (x^2 - 1) - 0 = x^2 - 1.
Therefore, the area of the cross-section at x is (x^2 - 1)^2.
To find the volume of the solid, we integrate the area of the cross-sections over the interval [x = -1, x = 2]:
V = ∫[(x^2 - 1)^2] dx
Evaluating this integral will give us the volume of the solid.
Let's solve the integral to find the volume of the solid.