Question

4. Find the volume of the solid whose base is bounded by the circle x2 + y2 = 4, with the cross sections taken perpendicular to the x-axis as equilateral triangles.

please draw a figure and round to 3 decimal places if needed

Answer 1

we are given

$x^2+y^2=4$

we can solve for y

y = 14-12

we can find side of triangle

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

now, we can find area of triangle

$A=\frac{\sqrt{3}}{4}s^2$

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

now, we can integrate area from -2 to 2 to get volume

$V=\int_{-2}^{2}\frac{\sqrt{3}}{4}(2\sqrt{4-x^2})^2dx$

$V=\int_{-2}^{2}\frac{\sqrt{3}}{4}4(4-x^2)dx$

$=\sqrt{3}\cdot \int _{-2}^24-x^2dx$

$=\sqrt{3}\left(\int _{-2}^24dx-\int _{-2}^2x^2dx\right)$

now, we can solve each integrals

and then combine them

$=\sqrt{3}\left(16-\frac{16}{3}\right)$

$V=18.475$...........Answer