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Evaluate the following double integrals.

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Input your answer as an integer or a fraction, not as a decimal.

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Two double integrals with numerical limits

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Input all numbers in your answer as integers or fractions, not as decimals.

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\$I = \\int^\\var{a}_{y=1} \\int^\\var{b}_{x=0} \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \$

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$I =$ [[0]]

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\$I = \\int^\\pi_{x=0} \\int^\\var{h}_{y=0} \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \$

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$I=$ [[0]]

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#### (a)

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We proceed to evaluate the double-integral:

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\\begin{align}
I &= \\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]_{x=0}^\\var{b} \\; \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right) \\; \\mathrm{d}y \\\\
&= \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1 \\; \\mathrm{d}y \\\\
&= \\simplify[std]{{c*b*a}+{d*b^2*a^2}-{c*b}-{d*b^2}} \\\\
&= \\simplify[std]{{(c*b*a)+(d*b^2*a^2)-(c*b)-(d*b^2)}}
\\end{align}

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#### (b)

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\\begin{align}
I &= \\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\\\
&= \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]_{y=0}^\\var{h} \\; \\mathrm{d}x \\\\[0.5em]
&= \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)} \\; \\mathrm{d}x  \\\\[0.5em]
&= \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0  \\\\[0.5em]
&= -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\[0.5em]
&= \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}}
\\end{align}

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