MATH 251 Lecture 33
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Surface Integrals
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is a finite surface.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} can be open (like a section of a surface) or closed (a surface that surrounds a solid)
Like with all integrals, we chop it up into pieces, and add up all the products of the area of the little piece and the value of the function:
- Parameterize the surface (a surface is defined by two parameters, usually Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} ) such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X(u,v) = (x(u,y), y(u,v), z(u,v))}
- Calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}S} , the "area element", in terms of parameters.
- Evaluate the resulting double integral.
Example: CoM of Hemisphere Shell
Find the center of mass of the upper hemisphere Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2+z^2=1} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \ge 0} . Assume that the mass density (per unit area) is constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \bar{x} &= 0 \\ \bar{y} &= 0 \\ \bar{z} &= \frac{1}{A} \iint_\Omega z \,\mathrm{d}S = \frac{1}{2\pi} \iint_\Omega z \,\mathrm{d}S \end{align}}
Parameterize the Region
We can use spherical coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1, \phi, \theta)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \le \tfrac{\pi}{2}} , so
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x(\phi,\theta) &= \sin{\phi}\cos{\theta} \\ y(\phi,\theta) &= \sin{\phi}\sin{\theta} \\ z(\phi,\theta) &= \cos{\phi} \end{align}}
Determine Area Element
For a rectangular region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega'} that maps to the hemisphere Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} , a small region gets mapped to a parallelogram.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{d}S &= \left\|\frac{\partial X}{\partial \phi} \times \frac{\partial X}{\partial \theta} \right\| \,\mathrm{d}\phi\,\mathrm{d}\theta \\ \frac{\partial X}{\partial \phi} &= \left\langle \cos{\phi}\cos{\theta}, \cos{\phi}\sin{\theta}, -\sin{\phi} \right\rangle \\ \frac{\partial X}{\partial \theta} &= \left\langle -\sin{\phi}\sin{\theta}, \sin{\phi}\cos{\theta}, 0 \right\rangle \end{align}}
Now we compute their cross product
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\phi} \begin{vmatrix}\hat\imath & \hat\jmath & \hat{k} \\ \cos{\phi}\cos{\theta} & \cos{\phi}\sin{\theta} & -\sin{\phi} \\ -\sin{\theta} & \cos{\theta} & 0 \end{vmatrix} = \sin{\phi} \left\langle \sin{\phi}\cos{\theta}, \sin{\phi}\sin{\theta}, \cos{\phi} \right\rangle}
Take the length of the resulting vector
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \sin{\phi} \right| \sqrt{\sin^2{\phi}\cos^2{\theta} + \sin^2{\phi}\sin^2{\theta} + \cos^2{\phi}} = \sin{\phi}}
since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \phi \le \tfrac{\pi}{2}} . Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}S = \sin{\phi}\,\mathrm{d}\phi\,\mathrm{d}\theta} .
Evaluate Double Integral
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} z(\phi,\theta) \sin{\theta}\,\mathrm{d}\phi\,\mathrm{d}\theta &= \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \cos{\phi} \sin{\theta}\,\mathrm{d}\phi\,\mathrm{d}\theta \\ &= \ldots = \pi \end{align}}
Multiply this value by the mass of the surface, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2\pi}} , so the final center of mass is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0,0,\tfrac{1}{2} \right)}
Example: Over Paraboloid
Set up the integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{\pi x}} over the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=x^2+y^2} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\le z \le 4} .
Parameterize the Region
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x &= u \\ y &= v \\ z &= u^2 + v^2 \end{align}}
Determine Area Element
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{d}S &= \left\| X_u \times X_v \right\| \mathrm{d}u\,\mathrm{d}v X_u &= \left\langle 1,0,2u \right\rangle \\ X_v &= \left\langle 1,0,2v \right\rangle \\ X_u \times X_v &= \left\langle -2u, -2v, 1 \right\rangle \\ \left\| X_u \times X_v \right\| &= \sqrt{4u^2 + 4v^2 + 1} \end{align}}
Set Up Integral
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_R \mathrm{e}^{\pi u} \sqrt{4u^2+4v^2+1} \,\mathrm{d}u\,\mathrm{d}v}
Round 2: Polar Coordinates
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x &= r\cos{\theta} \\ y &= r\sin{\theta} \\ z &= r^2 \\ X_r &= \left\langle \cos{\theta}, \sin{\theta}, 2r \right\rangle \\ X_{\theta} &= \left\langle -r\sin{\theta}, r\cos{\theta}, 0 \right\rangle \\ X_r \times X_{\theta} &= \left\langle -2r^2\cos{\theta}, -2r^2\sin{\theta}, r \right\rangle \\ \left\| X_r \times X_{\theta} \right\| &= r\sqrt{4r^2+1} \end{align}}
Therefore, the integral is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^{2\pi} \int_0^2 \mathrm{e}^{\pi r \cos{\theta}} r \sqrt{4r^2+1} \,\mathrm{d}r\,\mathrm{d}\theta}
For Functions of X and Y
given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z)} over a surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x,y)} :
Example
Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z) = yz} and the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y+z=4} in the first octant, find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_S f(x,y,z) \,\mathrm{d}S}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint_S yz \,\mathrm{d}S = \iint_A y(4-x-y) \sqrt{3} \,\mathrm{d}A}