Find the volume V of the described solid S

Question

The base of S is an elliptical region with boundary curve 4x^2 + 25y^2 = 100. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Answer 1

Solve for y:

 

y = 2√(25 - x²) / 5

 

The hypotenuse has length 4√(25 - x²) / 5 and is a side of a square composed of 4 of these sections as seen in this diagram (not to scale):

 

----------

|\        /|

|  \    /  |

|    \/    |

|    /\    |

|  /    \  |

|/        \|

----------

 

Thus the cross section's area is (1/4)[4√(25 - x²) / 5]² = 4(25 - x²)/25 = 4 - 4x²/25

 

The bounds on x are where the cross section has 0 area, which are at x = ±5.

 

Thus the volume is:

 

 

V = ∫_-5⁵ (4 - 4x²/25) dx = (4x - 4x³/75)|_-5⁵ = (20 - 20/3) - (-20 + 20/3) = 80/3 = 26.66666...

Answer 2

The ellipse intersects the x-axis at x = ±5.  A typical slice of the ellipse perpendicular to the x-axis has length 2y.  Since 4x²+25y²=100, y = (1/5)√(100-4x²). A typical cross section of the solid is a 45-45-90 triangle.  Since the hypotenuse has length 2y, Each leg of the right triangle has length √2y. The thickness of the cross section is Δx

 

Volume of cross section = (area of triangle)(thickness) = (1/2)(√2y)(√2y)Δx

 

                                                                                = y²Δx

 

                                                                                = (1/25)(100-4x²)Δx

 

Volume of solid = (1/25) ∫_{(from -5 to 5)}(100-4x²)dx 

 

                       = (2/25) ∫_{(from 0 to 5)}(100-4x²)dx (By symmetry)

 

                       = (2/25)[100x - (4/3)x³]_{(from 0 to 5)} = (2/25)[(500 - 166.66666) - 0] =

 

26 2/3 (or 80/3)