Question

If f(x,y)=6xy^2+3y^3 find (∫3,-2) f(x,y)dx.

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Frederik

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52 Answers

To evaluate the definite integral first evaluate the indefinite integral
$\int{ 6x{y}^{2}+3{y}^{3} } \mathrm{d} x$

Use the commutative property to reorder the terms
$\int{ 6{y}^{2}x+3{y}^{3} } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$
$\int{ 6{y}^{2}x } \mathrm{d} x+\int{ 3{y}^{3} } \mathrm{d} x$

Evaluate the indefinite integral
$3{x}^{2}{y}^{2}+\int{ 3{y}^{3} } \mathrm{d} x$

Evaluate the indefinite integral
$3{x}^{2}{y}^{2}+3x{y}^{3}$

To evaluate the definite integral, return the limits of integration
$\left. \left( 3{x}^{2}{y}^{2}+3x{y}^{3} \right) \right|_{ -2 }^{ 3 }$

Use $\left. F\left( x \right) \right|_{ a }^{ b }=F\left( b \right)-F\left( a \right)$ to evaluate the expression
$3 \times {3}^{2}{y}^{2}+3 \times 3{y}^{3}-\left( 3 \times {\left( -2 \right)}^{2} \times {y}^{2}+3 \times \left( -2 \right){y}^{3} \right)$

Simplify the expression
$15{y}^{2}+15{y}^{3}$

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