6.5 Average Value of a Function/2
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]\\[2ex]f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx\\[2ex]&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {(u)}{\frac {1}{4}}\,du={\frac {1}{8\pi }}\int _{-4\pi }^{4\pi }\sin(u)\,du&=-{\frac {1}{8\pi }}\cos(u){\bigg |}_{-4\pi }^{4\pi }&=-{\frac {1}{8\pi }}\cos(4\pi )+-{\frac {1}{8\pi }}\cos(-4\pi )\end{aligned}}}
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://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u &=4x \\[2ex] du &= 4dx \\[2ex] \frac{1}{4}du &= dx \end{align} }
New upper limit: 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://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\pi = 4(\pi)}
New lower limit:
