Question

Find the volumes of the solids whose bases are bounded by the circle with the indicated cross sections taken perpendicular to the axis

Answer 1

Explanation:

The missing information includes finding the volume of the circle x² + y² = 4 from the x-axis of the squares and the equilateral triangles.

`x^2 + y^2 = 4`

Perpendicular to x-axis

`y = √(4 -x^2)`

Area of the square =
`( √(4-x^2)+ √(4-x^2))^2`

`= (2 √(4 -x^2))^2 \\ \\ = 4(4-x^2) \\ \\ = 16 - 4x^2`

Volume V =
`\int ^2_(-2) (16 -4x^2) \ dx`

`= \Big [ 16x - (4x^3)/(3) \Big]^2_(-2)`

`= \Big [32 - (32)/(3) \Big] - \Big[-32 +(32)/(3) \Big]`

`= (128)/(3)`

Area of Equilateral triangle

`= (1)/(2)* 2 √(4-x^2)* √(3)√(4-x^2)`

`= √(3)(√(4-x^2))^2`

`Volume (V )= \int ^2_(-2)√(3) (4 -x^2) \ dx`

`= 4√(3) \ x - (√(3) \ x^3 )/(3) \Big|^2_(-2) \\ \\ = 16 √(3) - (16√(3))/(3) \\ \\ = (32√(2))/(3)`