You want to find the volume inside the hemisphere $x^2+y^2+z^2=4$ (i.e. inside the sphere but above the plane $z=0$ ) and outside the cylinder $x^2+y^2=1$ . Call this region $R$ .

In cylindrical coordinates, we have

$\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_1^2\int_0^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$

$\displaystyle=2\pi\int_1^2 r\sqrt{4-r^2}\,\mathrm dr$

$\displaystyle=-\pi\int_3^0\sqrt u\,\mathrm du$

(where $u=4-r^2$ )

$\displaystyle=\pi\int_0^3\sqrt u\,\mathrm du$

$=\dfrac{2\pi}3u^{3/2}\bigg|_0^3=2\sqrt3\,\pi$

7 0

3 years ago

How many ways can 6 people be seated in 4 seats

mojhsa [17]

switting upwards this should help

5 0

3 years ago

Help me soon due by 11 it 10:40!!!

Talja [164]

Answer:

1/4 divided by 1/3

1/3 divided into 1/4

How many 1/3 are in 1/4

Step-by-step explanation:

3 0

2 years ago

Read 2 more answers

There are 9 students in the school choir. The ratio of girls to boys in the choir is 5 : 4. Two girls are absent from practice o

pishuonlain [190]

Answer:

5:9

Step-by-step explanation:

4 0

3 years ago