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| | ::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math> | | ::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math> |
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| − | |Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: 0%">x = 1</math> and <math style="vertical-align: 0%">x = 4</math>. | + | |Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: -4%">x = 1</math> and <math style="vertical-align: -2%">x = 4</math>. |
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Revision as of 19:15, 15 May 2015
Find the area under the curve 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 y\,=\,\frac{8}{\sqrt{x}}}
between 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=1}
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 x=4}
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| Foundations:
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| For solving the problem, we only require the use of the power rule for integration:
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- 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 x^n dn = \frac{x^{n+1}}{n+1} + C.}
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| Geometrically, we need to integrate the region between the 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}
-axis, the curve, and the vertical lines 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 = 1}
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 x = 4}
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Solution:
| Step 1:
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| Set up the integral:
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- 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_1^{\,4} \frac{8}{\sqrt{x}} \,dx.}
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| Step 2:
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| Using the power rule we have:
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- 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{array}{rcl} \displaystyle{\int_1^{\,4} \frac{8}{\sqrt{x}}\,dx} & = & \displaystyle {\int_1^{\,4} 8x^{-1/2}\,dx}\\ \\ & = & \displaystyle{\frac{8 x^{1/2}}{2} \Bigr|_{x=1}^4}\\ \\ & = & 4x^{1/2} \Bigr|_{x=1}^4. \end{array}}
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| Step 3:
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| Now we need to evaluate to get:
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- 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 4x^{1/2} \Bigr|_{x=1}^4\,=\,4\cdot 4^{1/2} - 4\cdot 1^{1/2} \,=\, 8 - 4 \,=\, 4.}
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| Final Answer:
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- 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_1^{\,4} \frac{8}{\sqrt{x}} \,dx\,=\,4.}
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