Question

Find the volume of the solid bounded by z = 2 - x2 - y2 and z = 1. Express your answer as a decimal rounded to the hundredths place.

Answer 1

The answer is "
`((\pi)/(2))`".

Explanation:

`z = 2 - x^2 - y^2.........(1) \\\\z = 1.............(2)`

Let add equation 1 and 2:

Using formula:
`x^2+y^2=1 \\\\`

convert to polar coordinates

`r=2`

`\theta \varepsilon (0=\pi)=z\\\\V=\int^(2\pi)_(\theta=0)\int^(1)_(\pi=0)\int^{z_(2)}_(z_1) r \ dr \d \theta\\\\`

`=\int^(2\pi)_(0)\int^(1)_(0) (Z_2-z_1)r \ dr \d \theta\\\\=\int^(2\pi)_(0)\int^(1)_(0) 1- (-1-x^2-y^2) r \ dr \d \theta\\\\=\int^(2\pi)_(0)\int^(1)_(0) (+1 \pm r^2) r \ dr \d \theta\\\\=\int^(2\pi)_(0)\int^(1)_(0) (-r^3 + r) \ dr \d \theta\\\\=\int^(2\pi)_(0) (-(r^4)/(4)+(r^2)/(1))^(1)_(0) \d \theta\\\\=\int^(2\pi)_(0) ((1)/(4)) \d \theta\\\\=((2 \pi)/(4)) \\\\=((\pi)/(2)) \\\\`