6.1 Areas Between Curves/15

${\begin{aligned}&\color {red}\mathbf {y=\tan(x)} &\color {royalblue}\mathbf {y=2\sin(x)} \\&x=-{\frac {\pi }{3}}&x={\frac {\pi }{3}}\\\end{aligned}}$

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 \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left[(\tan(x)) - (2\sin(x))\right]dx }

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} \tan(x) &= 2\sin(x) \\ \tan(x)-2\sin(x) &= 0 \\ x &= 0 \\ \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_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left[(\tan(x)) - (2\sin(x))\right]dx = \int_{-\frac{\pi}{3}}^{0}\left[(\tan(x)) - (2\sin(x))\right]dx + \int_{0}^{\frac{\pi}{3}} \left[(2\sin(x)) - (\tan(x))\right]dx = \frac{14}{3} + \frac{64}{3} + \frac{14}{3} = \frac{92}{3}}

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_{-\frac{\pi}{3}}^{0}\left[(\tan(x)) - (2\sin(x))\right]dx \\[2ex] &= \left[\ln|sec(x)|+2\cos(x)\right]\Bigg|_{-\frac{\pi}{3}}^{0} \\[2ex] &= \left[\ln|sec(0)|+2\cos(0)\right]-\left[\ln|sec(-\frac{\pi}{3})+2\cos(-\frac{\pi}{3})|\right] \\[2ex] &= \left[0+2\right]-\left[\ln(2)-2(\frac{1}{2})\right] = -2\ln(2)-1 \\[2ex] &= -2\ln(2)-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 \begin{align} \int_{0}^{\frac{\pi}{3}} \left[(2\sin(x)) - (\tan(x))\right]dx \\[2ex] &= \left[-2\cos(x)-ln|sec(x)|\right]\Bigg|_{0}^{\frac{\pi}{3}} \\[2ex] &= \left[8(2)-\frac{2(2)^3}{3}\right] - \left[8(-2)-\frac{2(-2)^3}{3}\right] \\[2ex] &= \left[16-\frac{16}{3}\right]-\left[-16+\frac{16}{3}\right] = 32-\frac{32}{3} \\[2ex] &= \frac{64}{3} \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 \text{Note: } \int\tan(x)dx=\int\frac{\sin(x)}{\cos(x)}dx=\ln|\sec(x)|+C}

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 &= \cos(x) \\[2ex] du &= -\sin(x) \\[2ex] -du &= \sin(x)dx \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 (\frac{1}{u})dx &= -\ln|u|+C &=\ln|\cos(x)^{-1}|+C &=\ln|\frac{1}{cos(x)}|+C &=\ln|\sec(x)|+C \end{align} }