What is the integral from 0 to xy of 12x^2yz ?

The integral from 0 to xy of 12x^2yz is 6x^4y^3

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integral from 0 to xy of 12x^2yz

\int_{0}^{x y} 12 x^{2} yz d z

6 x^{4} y^{3}

Solution steps

\int_{0}^{x y} 12 x^{2} yz d z

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 x^{2} y \cdot \int_{0}^{x y} z d z

Apply the Power Rule

: [\frac{z ^{2}}{2}]_{0}^{x y}

= 12 x^{2} y [\frac{z ^{2}}{2}]_{0}^{x y}

Compute the boundaries:

\frac{x ^{2} y ^{2}}{2}

= 12 x^{2} y \frac{x ^{2} y ^{2}}{2}

Simplify

= 6 x^{4} y^{3}

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What is the integral from 0 to xy of 12x^2yz ?
The integral from 0 to xy of 12x^2yz is 6x^4y^3