the base of s is a elliptical region with boudary cuvrve 16x^2 +16y^2 =4. cross sections perpandicular to the x axis are isosceles right triangles with hypotenuse in the base. find the volume of s
To find the volume of the solid (S), we first need to determine the limits of integration for the base of S.
Given that the base is an elliptical region with the boundary curve 16x^2 + 16y^2 = 4, we can rewrite this equation as x^2 + y^2 = 1/4.
Since the cross sections of S are perpendicular to the x-axis and take the shape of isosceles right triangles with the hypotenuse in the base, it means that the base has rotational symmetry around the x-axis.
To find the limits of integration for the base, we need to find the points where the ellipse intersects the x-axis.
Setting y = 0 in the equation x^2 + y^2 = 1/4, we have:
x^2 + 0 = 1/4
x^2 = 1/4
x = ±(1/2)
So, the limits of integration for the base of S on the x-axis are x = -1/2 and x = 1/2.
Now, let's consider a vertical slice of S at a specific x-value. This slice will be an isosceles right triangle with legs of length 2y and a hypotenuse in the base.
The length of the legs, 2y, will vary depending on the x-value. We can express y in terms of x by using the equation x^2 + y^2 = 1/4:
y^2 = 1/4 - x^2
y = √(1/4 - x^2)
The area of this isosceles right triangle slice can be found using the formula for the area of a right triangle: Area = (1/2) * base * height.
The base of the triangle is the length of the hypotenuse in the base, which corresponds to the length of the major axis of the ellipse. It is given by 2y, which we already found above.
The height of the triangle can be determined by subtracting the y-coordinate of the ellipse at that particular x-value from the y-coordinate at the x-axis (y = 0). Since the ellipse is symmetric around the y-axis, the y-coordinate at the x-axis is the same as the magnitude of the y-coordinate at any other point on the ellipse.
Therefore, the height of the triangle is given by h = |y| = √(1/4 - x^2).
Now, we can calculate the volume of S by integrating the areas of these isosceles right triangle slices over the range of x-values from -1/2 to 1/2.
Volume = ∫[from -1/2 to 1/2] (1/2) * (2y) * (√(1/4 - x^2)) dx
Simplifying the equation and integrating will give us the volume of S.