ii. find the volume of the solid formed when ( y = - ( x ^ { 2 } - 4 ) ) is rotated about the ( y - ) axis.

Explanation:

Step1: Express (x) in terms of (y)

Given (y =-(x^{2}-4)), then (x^{2}=4 - y), so (x=\pm\sqrt{4 - y}). Since we are dealing with volume, we can use the formula for the volume of a solid of revolution about the (y) - axis (V=\pi\int_{a}^{b}[x]^{2}dy). Here, when (x = 0), (y = 4) (lower limit (a = 0), upper limit (b = 4)) and ([x]^{2}=4 - y).

Step2: Integrate the function

[ \begin{align*} V&=\pi\int_{0}^{4}(4 - y)dy\ &=\pi\left(4y-\frac{y^{2}}{2}\right)\big|_{0}^{4}\ \end{align*} ]

Step3: Evaluate the definite integral

[ \begin{align*} V&=\pi\left[\left(4\times4-\frac{4^{2}}{2}\right)-\left(4\times0-\frac{0^{2}}{2}\right)\right]\ &=\pi\left(16 - 8-0\right)\ &=8\pi \end{align*} ]

Answer:

(8\pi)