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 use change of variable, $\int _{0}^{1}\int _{0}^{x}e^{\frac {y}{x}}~dydx=\int _{0}^{1}[xe^{\frac {y}{x}}|_{y=0}^{y=x}]~dx$