Find the average value of the function on the given interval.
![{\displaystyle f(x)=2x^{3}(1+x^{2})^{4},~~~[0,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a161ffdc7696e36689931b0c93c832f51cbe78)
| Foundations:
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| The average value of 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)}
on an interval 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 [a,b]}
is given 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 f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)dx}
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Solution:
| Step 1:
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| Using the formula given in the Foundations sections, 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 f_{\text{avg}}=\frac{1}{2-0}\int_0^2 2x^3(1+x^2)^4dx=\int_0^2 x^3(1+x^2)^4dx}
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| Step 2:
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| 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 (1+x^2)^2=1+2x^2+x^4}
, 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 (1+x^2)^4=(1+x^2)^2(1+x^2)^2=(1+2x^2+x^4)(1+2x^2+x^4)=1+4x^2+6x^4+4x^6+x^8}
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| So, the integral becomes 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_{\text{avg}}=\int_0^2 x^3(1+4x^2+6x^4+4x^6+x^8)dx}
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| Step 3:
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| We distribute 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 f_{\text{avg}}=\int_0^2 x^3(1+4x^2+6x^4+4x^6+x^8)dx=\int_0^2 (x^3+4x^5+6x^7+4x^9+x^{11})dx}
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| Now, we integrate 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 f_{\text{avg}}=\left.\frac{x^4}{4}+4\frac{2x^6}{3}+\frac{3x^8}{4}+\frac{2x^{10}}{5}+\frac{x^{12}}{12}\right|_{0}^{2}}
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