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

Which function is the inverse of f(x)=9x-3

DIA [1.3K]

F^-1(x)= x/9 + 1/3

To find this, just interchange the variables and solve for y.

y=9x-3
x=9y-3
x+3=9y
divide by nine

3 0

3 years ago

What is number 61 and a step by step explaination please?

astra-53 [7]

1/3 ln(x) + ln(2) - ln(3) = 3

Recall that $m\log_b(n)=\log_b(n^m)$ , so

ln(x ¹ʹ³) + ln(2) - ln(3) = 3

Condense the left side by using sum and difference properties of logarithms:

$\log_b(m)+\log_b(n)=\log_b(mn)$

$\log_b(m)-\log_b(n)=\log_b\left(\dfrac mn\right)$

Then

ln(2/3 x ¹ʹ³) = 3

Take the exponential of both sides; that is, write both sides as powers of the constant e. (I'm using exp(x) = e ˣ so I can write it all in one line.)

exp(ln(2/3 x ¹ʹ³)) = exp(3)

Now exp(ln(x)) = x for all x, so this simplifies to

2/3 x ¹ʹ³ = exp(3)

Now solve for x. Multiply both sides by 3/2 :

3/2 × 2/3 x ¹ʹ³ = 3/2 exp(3)

x ¹ʹ³ = 3/2 exp(3)

Raise both sides to the power of 3:

(x ¹ʹ³)³ = (3/2 exp(3))³

x = 3³/2³ exp(3×3)

x = 27/8 exp(9)

which is the same as

x = 27/8 e ⁹

3 0

3 years ago

The scatter plot shows the number of hats and scarves each knitter sold at a knitting show.

SOVA2 [1]

I believe your answer should be 5!

4 0

3 years ago

Read 2 more answers