MATH 251 Lecture 24

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Triple Integrals

Given a 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 R \in \mathbb{R}^3 = [a,b] \times [c,d] \times [e,f]} ,

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 \iiint_R f(x,y,z) \,\mathrm{d}V = \int_a^b \int_c^d \int_e^f f(x,y,z) \,\mathrm{d}z \,\mathrm{d}y \,\mathrm{d}x}

Example with Simplification

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 R = [-1,1] \times [0,2] \times [0,1]}

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 f(x,y,z) = x\sin{y} + \mathrm{e}^z}

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_0^1 \int_0^2 \int_{-1}^1 \left(x\sin{y} + \mathrm{e}^z \right) \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z}

The integral can be split into a sum of two integrals:

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_0^1 \int_0^2 \int_{-1}^1 x\sin{y} \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z + \int_0^1 \int_0^2 \int_{-1}^1 \mathrm{e}^z \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z}

The first integral is over an odd region with respect to x, so for every point on one side, there is an equal and opposite point on the other side, so the entire integral will evaluate to 0:

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_0^1 \int_0^2 \int_{-1}^1 \mathrm{e}^z \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z}

The function inside this integral does not depend on x or y, so we can multiply the remaining integral with respect to z by the lengths of the intervals along x and y:

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 (2-0)(1-(-1)) \int_0^1 \mathrm{e}^z \,\mathrm{d}z = 4\int_0^1 \mathrm{e}^z \,\mathrm{d}z}

This is easy to evaluate as normal:

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 4(\mathrm{e}-1)}

Triple Integrals in Spherical Coordinates

Find the center of mass of a unit 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} , 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} . Density is a 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(x,y,z) = \rho_0}

Find the mass: 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{2}{3} \pi \rho_0} ... that was easy.

Due to symmentry, 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 \bar{x} = \bar{y} = 0} .

Therefore, we only need to 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 \bar{z} = \frac{3}{2\pi\rho_0} \iiint_R z\rho_0 \,\mathrm{d}V = \frac{3}{2\pi} \iiint_R z\,\mathrm{d}V}

This integral would be easier to evaluate by converting to cylindrical (polar) coordinates 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 z = \sqrt{1-x^2-y^2} = \sqrt{1-r^2}}

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 \frac{3}{2\pi} \int_0^{2\pi} \int_0^1 \frac{1}{2}\left(1-r^3 \right) r^3 \,\mathrm{d}r \,\mathrm{d}\theta = \ldots = \frac{3}{8}}

Another Example

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) = x^2-3xy-2z} , 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 R} is bounded by 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+2y+3z \le 5} in the first octant.

Let's evaluate in the order 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}y \mathrm{d}z \mathrm{d}x}

Find the bounds:

  • 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 y \le \tfrac{1}{2}(6-x-3z)}
  • 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 \tfrac{1}{3}(6-x)} (plug in 0 for 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 y} )
  • 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 x \le 6} (plug in 0 for 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} 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 y} )

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_0^6 \int_0^{\tfrac{1}{3}(6-x)} \int_0^{\tfrac{1}{2}(6-x-3z)} (x^2 + 3xy-2z) \,\mathrm{d}y \,\mathrm{d}z \,\mathrm{d}x}

And due to time constraints, we will not evaluate that integral.

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