Multivariate Calculus 10B, Problem 1

Calculate the following integrals

a) 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 \int _0^1 \int_y^1 e^{\frac{y}{x}}~dxdy}

b) 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 \int_0^1 \int_0^{cos^{-1}(y)} e^{2x-y}~dxdy}

solution(a):

Here we change order of integration, $\int _{0}^{1}\int _{0}^{x}e^{\frac {y}{x}}~dydx=\int _{0}^{1}[xe^{\frac {y}{x}}|_{y=0}^{y=x}]~dx=\int _{0}^{1}x(e-1)~dx={\frac {1}{2}}x^{2}|_{0}^{1}(e-1)={\frac {1}{2}}(e-1)$ solution(b): Here we change order of integration, 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 \int _0^{\frac{\pi}{2}} \int_0^{cos(x)} e^{2x - y}~dydx = \int _0^{\frac{\pi}{2}}[-e^{2x -y}|_{y = 0}^{y = cos(x)}]~dx = \int_0^{\frac{\pi}{2}} [e^{2x} - e^{2x - cos(x)}]~dx = [\frac{1}{2}e^{2x} - \frac{1}{2 + sin(x)}e^{2x - cos(x)}]|_0^{\frac{\pi}{2}} = \frac{e^{\pi}}{6} + \frac{1}{2}(\frac{1}{e} - 1)}