Determine the mass of the lamina in the first quadrant bounded by the coordinate axes and the...

Question:

Determine the mass of the lamina in the first quadrant bounded by the coordinate axes and the curve {eq}y=e^{-8x} {/eq} if the density function {eq}\delta(x,y)=xy {/eq}.

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Answer and Explanation:

Integration:

A mathematical quantity that represents the joining of the infinitesimal form of the derivative function to analyse the continuous variation of the function is known as integration. It used in engineering applications to analyse the system.


Given Data:

  • The curve is: {eq}y = {e^{ - 8x}} {/eq}
  • The density function is: {eq}\delta \left( {x,y} \right) = xy {/eq}


The expression for mass of lamina is

{eq}dM = \delta \left( {x,y} \right)dydx {/eq}


Integrate the above expression with following limits

{eq}\begin{align*} 0 &\le y \le {e^{ - 8x}}\\ 0 &\le x \le \infty \end{align*} {/eq}

{eq}\begin{align*} \int {dM} &= \int_0^\infty {\int_0^{{e^{ - 8x}}} {xydydx} } \\ M &= \int_0^\infty {\left[ {\dfrac{{{y^2}}}{2}} \right]_0^{{e^{ - 8x}}}x} dx\\ &= \int_0^\infty {\left[ {\dfrac{{{{\left( {{e^{ - 8x}}} \right)}^2}}}{2} - 0} \right]} xdx\\ &= \dfrac{1}{2}\int_0^\infty {{e^{ - 16x}}} xdx \cdots\cdots\cdots{\rm(I)} \end{align*} {/eq}


The definition of gamma function the expression

{eq}\int_0^\infty {{e^{ - cx}}{x^{n - 1}}dx} = \dfrac{{n!}}{{{c^n}}} {/eq}


Solve the expression (I) by gamma function

{eq}\begin{align*} M &= \dfrac{1}{2}\int_0^\infty {{e^{ - 16x}}} xdx\\ &= \dfrac{1}{2}\dfrac{{2!}}{{{{\left( {16} \right)}^2}}}\\ &= \dfrac{2}{{2\left( {256} \right)}}\\ &= \dfrac{1}{{256}} \end{align*} {/eq}


Thus the mass of lamina is {eq}\dfrac{1}{{256}} {/eq}


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Basic Calculus: Rules & Formulas

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Chapter 3 / Lesson 6
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