Question

Which of the following is the volume of the shaded in Of the sphere?

Answer 1

The correct option is;

πR³(2/3 + (1/3)cos³θ - cosθ)

Step-by-step explanation:

The volume of a segment of a sphere is given by the relation;

`V = \pi \cdot h^2 \cdot \left (R - (h)/(3) \right)`

We note that h = R - R·cos(θ)

Therefore by substituting the value of h in the equation of a segment of a sphere, we have;

`V = \pi \cdot \left (R - R\cdot cos(\theta) \right ) ^2 \cdot \left (R - (\left (R - R\cdot cos(\theta) \right ))/(3) \right)`

Which gives;

`(R^3\cdot \pi \cdot cos^3 (\theta) -3 \cdot R^3 \cdot\pi \cdot cos (\theta) + 2 \cdot R^3 \cdot \pi)/(3)`

`R^3\cdot \pi \cdot \left ((cos^3 (\theta) -3 \cdot cos (\theta) + 2 )/(3) \right )`

`R^3\cdot \pi \cdot \left ((cos^3 (\theta) + 2 )/(3) - cos (\theta) \right )`

`R^3\cdot \pi \cdot \left ((cos^3 (\theta) )/(3) + (2)/(3) - cos (\theta) \right )`

Therefore, the correct option is πR³(2/3 + (1/3)cos³θ - cosθ).