R is the region in the plane bounded below by the curve y=x^2 and above by the line y=1.
(a) Set up and evaluate an integral that gives the area of R.
(b) A solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Find the volume of the solid.
(c) A solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Find the volume of the solid.
To find the area of region R, we need to determine the limit of integration and set up the integral. Here's how to do it:
(a) The region R is bounded below by the curve y = x^2 and above by the line y = 1. To find the limit of integration, we need to determine the x-values at which the two curves intersect.
Setting the equations y = x^2 and y = 1 equal to each other:
x^2 = 1
Taking the square root of both sides, we get:
x = ±1
So the limits of integration are x = -1 to x = 1.
The integral to find the area of R is then given by:
A = ∫[-1,1] (1 - x^2) dx
To evaluate this integral, we simply integrate the function (1 - x^2) with respect to x between the limits -1 and 1.
(b) To find the volume of the solid with square cross-sections, we need to consider the height of each square at different y-values within region R.
The height of each square is given by the difference between the upper curve y = 1 and the lower curve y = x^2.
To calculate the volume, we integrate the area of the square cross-sections with respect to y within the limits of y = 0 to y = 1 (since R is bounded between y = 0 and y = 1).
The integral to find the volume is then given by:
V = ∫[0,1] [(1 - x^2)^2] dy
(c) To find the volume of the solid with equilateral triangle cross-sections, we need to determine the height of each equilateral triangle at different y-values within region R.
The height of each equilateral triangle is given by the difference between the upper curve y = 1 and the lower curve y = x^2.
Since the height of an equilateral triangle is given by h = √3/2 * s (where s represents the side length), and for each y-value within R, the side length of the triangle is determined by the difference between the x-values at which the y-value intersects the curves y = 1 and y = x^2, we can calculate the volume of the solid.
The integral to find the volume is then given by:
V = ∫[0,1] [(√3/2 * (1 - x^2))^2] dy