Find the volume of the solid whose base is the region enclosed by y=x^2 and y=2, and the cross sections perpendicular to the y-axis are squares.
To find the volume of the given solid, we need to use the method of cross-sectional areas.
First, let's find the limits of integration. We know that the base of the solid is the region enclosed by the curves y = x^2 and y = 2. To find the limits along the y-axis, we need to determine the values of y where the two curves intersect.
Setting the two equations equal to each other:
x^2 = 2
x = +/- sqrt(2)
Since we are interested in the positive region, the limits for x are sqrt(2) to -sqrt(2).
Now, let's consider a generic cross-sectional area at a height y.
The width of the square cross section at height y will be the difference between the x-values of the points where the curve y = x^2 intersects the line y = 2.
Given y = 2, the equation becomes:
2 = x^2
x^2 - 2 = 0
Solving this equation, we get:
x = +/- sqrt(2)
Again, we consider the positive region, so the x-values for y = 2 are sqrt(2) and -sqrt(2). Thus, the width of the square cross section at height y will be 2 * sqrt(2).
The area of a square cross section is simply the square of the side length, which in this case is 2 * sqrt(2).
The volume of the solid can be obtained by integrating the cross-sectional areas over the range of y.
∫[sqrt(2), 2](2 * sqrt(2))^2 dy
Simplifying, we have:
∫[sqrt(2), 2] 8 dy
Now, integrating, we get:
8y | [sqrt(2), 2]
Substituting the limits, we have:
8(2) - 8(sqrt(2))
Simplifying further, we get:
16 - 8(sqrt(2))
So, the volume of the solid is 16 - 8(sqrt(2)) cubic units.
To find the volume of the solid, we can use the method of integration.
First, let's find the limits of integration for y. The region enclosed by y = x^2 and y = 2 is bounded by the x-axis and the parabola. To find the x-values that define this region, set the two equations equal to each other:
x^2 = 2
Solving for x, we get x = ±√2. Since we're interested in the region enclosed by the curves, we'll only consider the positive value, so x ranges from -√2 to √2.
Next, let's consider a cross-section of the solid perpendicular to the y-axis. We're told that these cross-sections are squares. Since the squares are perpendicular to the y-axis, the sides of each square will have lengths equal to the difference in y-values at the same x-coordinate.
In this case, the difference in y-values is given by y = 2 - x^2. Thus, the side length of each square is 2 - x^2.
Finally, the volume of the solid can be computed by integrating the area of each cross-section from the lower bound to the upper bound of x:
V = ∫[lower bound, upper bound] A(x) dx
where A(x) is the area of the cross-section at the given x-coordinate.
Since the cross-sections are squares, the area of each cross-section is (side length)^2 which is (2 - x^2)^2.
Therefore, the volume can be calculated as:
V = ∫[-√2, √2] (2 - x^2)^2 dx
For some value of y, the base of the square is 2x, so its area is 4x^2 = 4y.
So, add up all those thin square sheets of thickness dy, and you get
v = ∫[0,2] 4y dy = 8