6.1 Areas Between Curves/15
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} & \color{red}\mathbf{y=\tan(x)} & \color{royalblue}\mathbf{y= 2\sin(x)} \\ & x=-\frac{\pi}{3} & x=\frac{\pi}{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 \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|secx|+2\cos(x)\right]\Bigg|_{-\frac{\pi}{3}}^{0} \\[2ex] &= \left[\frac{2(-2)^3}{3}-8(-2)\right]-\left[\frac{2(-3)^3}{3}-8(-3)\right] \\[2ex] &= \left[\frac{-16}{3}+16\right]-\left[\frac{-54}{3}+24\right] = \frac{38}{3}-8 \\[2ex] &= \frac{14}{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 \begin{align} \int_{-2}^{2} \left((8-x^2) - (x^2)\right)dx &= \int_{-2}^{2}\left(8-2x^2\right)dx \\[2ex] &= \left[8x-\frac{2x^3}{3}\right]\Bigg|_{-2}^{2} \\[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} }