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

Answer:

Step-by-step 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

$y = \sqrt{4 -x^2}$

Area of the square = $( \sqrt{4-x^2}+ \sqrt{4-x^2})^2$

$= (2 \sqrt{4 -x^2})^2 \\ \\ = 4(4-x^2) \\ \\ = 16 - 4x^2$

Volume  V = $\int ^2_{-2} (16 -4x^2) \ dx$

$= \Big [ 16x - \dfrac{4x^3}{3} \Big]^2_{-2}$

$= \Big [32 - \dfrac{32}{3} \Big] - \Big[-32 +\dfrac{32}{3} \Big]$

$= \dfrac{128}{3}$

Area of Equilateral triangle

$= \dfrac{1}{2}\times 2 \sqrt{4-x^2}\times \sqrt{3}\sqrt{4-x^2}$

$= \sqrt{3}(\sqrt{4-x^2})^2$

$Volume (V )= \int ^2_{-2}\sqrt{3} (4 -x^2) \ dx$

$= 4\sqrt{3} \ x - \dfrac{\sqrt{3} \ x^3 }{3} \Big|^2_{-2} \\ \\ = 16 \sqrt{3} - \dfrac{16\sqrt{3}}{3} \\ \\ = \dfrac{32\sqrt{2}}{3}$