Please help! Show work and answers! The base of a certain solid is an elliptical
ID: 2829741 • Letter: P
Question
Please help! Show work and answers!
The base of a certain solid is an elliptical region with boundary curve 16x2+25y2=400. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
Use the formula V=?baA(x)dx to find the volume of the solid.
The lower limit of integration is a =
The upper limit of integration is b =
The base of the triangular cross-section is the following function of x:
The height of the triangular cross-section is the following function of x:
The area of the triangular cross-section is A(x)=
Thus the volume of the solid is V =
Explanation / Answer
x^2/25 + y^2/16 = 1
y^2/16 = 1 - x^2/25
y^2 = 16(1 - x^2/25)
The area of a cross-sectional slice is (1/2)(2y)^2 = 2y^2 = 2(16)(1 - x^2/25)
dV = 32(1 - x^2/25) dx
The ellipse stretches from x = -5 to x = 5, but we can use symmetry to make it easier to calculate the volume.
(1/2) V = 32 ? (1 - x^2/25) dx {from x = 0 to x = 5} = 32 (x - x^3/75)
V = 64 (x - x^3/75) ] {0, 5} = 64 [(5 - 5^3/75) - (0 - 0^3/3)] ) = 213.33