Respuesta :
Answer:
The area of surface is [tex]\dfrac{2}{3}(10^{\frac{3}{2}}-1)\pi[/tex]
Step-by-step explanation:
Given that,
The equation of cylinder is
[tex]x^2+y^2=9[/tex]
The part of the surface z = xy
The coordinates is,
[tex]z_{x}=y[/tex]
[tex]z_{y}=x[/tex]
We need to calculate the value of ds
Using formula of ds
[tex]ds=\sqrt{1+z_{x}^2+z_{y}^2}dA[/tex]
Put the value in to the formula
[tex]ds=\sqrt{1+y^2+x^2}dA[/tex]....(I)
We know that.
The polar coordinates,
[tex]x=r\cos\theta[/tex]
[tex]y=r\sin\theta[/tex]
The general equation of cylinder is
[tex]x^2+y^2=r^2[/tex]
compare from given equation
[tex]x^2+y^2=3^2[/tex]
0<=θ<=2π, 0<=r<=3
Area element in polar coordinates is,
[tex]dA=r dr d\theta[/tex]
Put the value of dA in equation (I)
[tex]ds=\sqrt{1+r^2}r dr d\theta[/tex]....(II)
We need to calculate the area of surface
Using equation (II)
[tex]s=\int_{0}^{2\pi}\int_{0}^{3}\sqrt{1+r^2}r dr d\theta[/tex]
[tex]s=\int_{0}^{2\pi}\dfrac{1}{3}((1+r^2)^{\frac{3}{2}})_{0}^{3}d\theta[/tex]
[tex]s=\int_{0}^{2\pi}\dfrac{1}{3}((1+3^2)^{\frac{3}{2}}-(1+0^2)^{\frac{3}{2}})d\theta[/tex]
[tex]s=(\dfrac{1}{3}((10)^{\frac{3}{2}}-1)\theta)_{0}^{2\pi}[/tex]
[tex]s=\dfrac{1}{3}(10^{\frac{3}{2}}-1)(2\pi-0)[/tex]
[tex]s=\dfrac{2}{3}(10^{\frac{3}{2}}-1)\pi[/tex]
Hence, The area of surface is [tex]\dfrac{2}{3}(10^{\frac{3}{2}}-1)\pi[/tex]
