Due to the symmetry of the paraboloid about the <em>z</em>-axis, you can treat this is a surface of revolution. Consider the curve $y=x^2$ , with $1\le x\le2$ , and revolve it about the <em>y</em>-axis. The area of the resulting surface is then

$\displaystyle2\pi\int_1^2x\sqrt{1+(y')^2}\,\mathrm dx=2\pi\int_1^2x\sqrt{1+4x^2}\,\mathrm dx=\frac{(17^{3/2}-5^{3/2})\pi}6$

But perhaps you'd like the surface integral treatment. Parameterize the surface by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+u^2\,\vec k$

with $1\le u\le2$ and $0\le v\le2\pi$ , where the third component follows from

$z=x^2+y^2=(u\cos v)^2+(u\sin v)^2=u^2$

Take the normal vector to the surface to be

$\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial u}=-2u^2\cos v\,\vec\imath-2u^2\sin v\,\vec\jmath+u\,\vec k$

The precise order of the partial derivatives doesn't matter, because we're ultimately interested in the magnitude of the cross product:

$\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right\|=u\sqrt{1+4u^2}$

Then the area of the surface is

$\displaystyle\int_0^{2\pi}\int_1^2\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right\|\,\mathrm du\,\mathrm dv=\int_0^{2\pi}\int_1^2u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv$

which reduces to the integral used in the surface-of-revolution setup.

7 0

3 years ago

A bag contains green marbles and red marbles, 43 in total. The number of green

sukhopar [10]

26

r + 2r - 8 = 43
3r = 51
r = 17
g = 2r-8 = 34-8=26

Mark brainliest please

6 0

3 years ago

ΔACB • ΔEFD

Vaselesa [24]

Answer:

y=4

Step-by-step explanation:

8 0

3 years ago

A car salesman earns a 20% commission on every car that he sells. Approximately what commission would he make if he sold a car f

monitta

First you are going to multiply the number by the percent. 9,895 x 20= 197,900.

Now divide the answer by 100. 197,900 / 100= 1,979.

This means the car salesman should make 1,979 from the car sale.

Remember, multiply the number by the percent, then divide that answer by 100.

I hope this helps! :)

6 0

3 years ago