Find the volume of the solid S that satisfies the two following conditions. First, the base of S is the elliptical region with boundary curve 9 x2 + 4 y2 = 36, and second, the cross-sections of S perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
To find the volume of the solid S, we need to determine the limits of integration and the formula to calculate the volume.
First, let's consider the base of S, which is the elliptical region with the boundary curve 9x^2 + 4y^2 = 36. This equation represents an ellipse centered at the origin with semi-major axis of length 3 in the x-direction and semi-minor axis of length 2 in the y-direction. To find the limits of integration in the y-direction, we can solve this equation for y.
9x^2 + 4y^2 = 36
Divide both sides by 36:
x^2/4 + y^2/9 = 1
Solving for y:
y = ±3√(1 - x^2/4)
Therefore, the limits of integration in the y-direction are y = -3√(1 - x^2/4) to y = 3√(1 - x^2/4).
Now, let's consider the cross-sections of S perpendicular to the x-axis. According to the problem statement, these cross-sections are isosceles right triangles with the hypotenuse in the base. In an isosceles right triangle, the length of the two equal sides is equal to the length of the hypotenuse. Therefore, the length of the two legs of each isosceles right triangle is equal to the distance between the top and bottom curves of the base ellipse at the same x-value.
To find this distance, we can subtract the two curves:
h = 3√(1 - x^2/4) - (-3√(1 - x^2/4))
Simplifying, we have:
h = 6√(1 - x^2/4)
The volume of a solid with cross-sections perpendicular to the x-axis can be calculated using the integral:
V = ∫[a,b] A(x) dx
where A(x) represents the area of the cross-section at a given x-value, and [a, b] are the limits of integration in the x-direction.
In this case, the area of each cross-section is given by:
A(x) = 0.5 * base * height
A(x) = 0.5 * (6√(1 - x^2/4))^2
Simplifying, we have:
A(x) = 0.5 * 36 * (1 - x^2/4)
A(x) = 18 - 4.5x^2
Therefore, the volume of the solid S can be calculated as:
V = ∫[-2, 2] (18 - 4.5x^2) dx
Integrating, we have:
V = [18x - 1.5x^3/3] evaluated from -2 to 2
V = (18(2) - 1.5(2)^3/3) - (18(-2) - 1.5(-2)^3/3)
V = (36 - 16/3) - (-36 + 16/3)
V = (108/3 - 16/3) - (-108/3 + 16/3)
V = 92/3 - (-92/3)
V = 184/3
Therefore, the volume of the solid S is 184/3 cubic units.