6.1 Areas Between Curves/22: Difference between revisions
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<math> | <math> | ||
\sin(\frac{\ | \begin{align} | ||
x= 0 | \sin(\frac{x\pi}{2}) &= x \\ | ||
x=1 | x &= 0 \\ | ||
x &=1 \\ | |||
\end{align} | |||
</math> | |||
<math> | |||
\int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right) - x\right)dx = \int_{0}^{1}\left(\sin\left(\frac{\pi}{2}\right)\right)dx - \int_{0}^{1} (x)dx = \frac{2}{\pi} - \frac{1}{2} | |||
</math> | |||
<math> | |||
\begin{align} | |||
\int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right)\right)dx \\ | |||
u = \frac{x\pi}{2} \\ | |||
du = \frac{\pi}{2}dx \\ | |||
\frac{2}{\pi}du =dx \\ | |||
\end{align} | |||
</math> | |||
New upper limit: <math>\frac{(0)\pi}{2}=0 </math> | |||
New lower limit: <math>\frac{(1)\pi}{2} = \frac{\pi}{2} </math> | |||
<math> | |||
\begin{align} | |||
\int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right)\right) dx &= \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \sin(u) du \\ | |||
&= \frac{2}{\pi} \left[-\cos(u)\right]\Bigg|_{0}^{\frac{\pi}{2}} \\ | |||
&= \frac{2}{\pi} \left[-\cos(\frac{\pi}{2})+\cos(0)\right] \\ | |||
&= \frac{2}{\pi} [0+1] = \frac{2}{\pi} \\ | |||
\end{align} | |||
</math> | |||
<math> | |||
\begin{align} | |||
\int_{0}^{1} x dx &= \left[\frac{x^2}{2}\right]\Bigg|_{0}^{1} \\ | |||
&= \frac{1}{2} - 0 = \frac{1}{2} \\ | |||
\end{align} | |||
</math> | </math> | ||
Latest revision as of 03:29, 20 September 2022
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}&\color {red}\mathbf {y=\sin({\frac {\pi x}{2}})} &\color {royalblue}\mathbf {y=x} \\\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} \sin(\frac{x\pi}{2}) &= x \\ x &= 0 \\ x &=1 \\ \end{align} }
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 \int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right) - x\right)dx = \int_{0}^{1}\left(\sin\left(\frac{\pi}{2}\right)\right)dx - \int_{0}^{1} (x)dx = \frac{2}{\pi} - \frac{1}{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://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right)\right)dx \\ u = \frac{x\pi}{2} \\ du = \frac{\pi}{2}dx \\ \frac{2}{\pi}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 \frac{(0)\pi}{2}=0 }
New lower 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 \frac{(1)\pi}{2} = \frac{\pi}{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://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{0}^{1} \left(\sin\left(\frac{x\pi}{2}\right)\right) dx &= \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \sin(u) du \\ &= \frac{2}{\pi} \left[-\cos(u)\right]\Bigg|_{0}^{\frac{\pi}{2}} \\ &= \frac{2}{\pi} \left[-\cos(\frac{\pi}{2})+\cos(0)\right] \\ &= \frac{2}{\pi} [0+1] = \frac{2}{\pi} \\ \end{align} }
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} \int_{0}^{1} x dx &= \left[\frac{x^2}{2}\right]\Bigg|_{0}^{1} \\ &= \frac{1}{2} - 0 = \frac{1}{2} \\ \end{align} }