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SHOW ALL WORK!!!
Problem 1 [6 points] Use the general slicing method to find the volume of the following
solid. The solid whose base is the region bounded by the ellipse
x
2
y
2
+
17 8
= 1 and whose
cross sections perpendicular to the base and parallel to the x-axis are semicircles.
Solution:
To find the volume of the solid, we need to use the general slicing
method, which involves integrating the area of the cross sections along
the axis of symmetry. In this case, the axis of symmetry is the x-axis,
and the cross sections are semicircles. The area of a semicircle with
radius r is A = (1/2)Ï€r^2. To find the radius of each semicircle, we
need to use the equation of the ellipse, x^2/17 + y^2/8 = 1, and solve
for y in terms of x. We get y = ±√(8 - 8x^2/17). Since we only need the
positive value of y, we can take the positive square root as the radius.
Therefore, the area of each cross section is A = (1/2)π(√(8 -
8x^2/17))^2 = 4Ï€(1 - x^2/17). To find the limits of integration, we
need to find where the ellipse intersects the x-axis. This happens when
y = 0, so we get x = ±√17. Therefore, the volume of the solid is V =
∫[A(x)dx] from -√17 to √17. Substituting A(x) with 4π(1 - x^2/17), we
get V = ∫[4π(1 - x^2/17)dx] from -√17 to √17. Using the power rule and
evaluating at the limits, we get V = [4π(x - x^3/51)] from -√17 to √17.
Simplifying and subtracting, we get V = 16π√17/3 - 32π√17/51 ≈ 28.64
cubic units.
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