# Calculate the triple integral over E of x^2 e^y dV where E is bounded by the parabolic cylinder z...

## Question:

Calculate {eq}\displaystyle\iiint_{E} x^2 e^y \, \mathrm{d}V \; {/eq} where {eq}E {/eq} is bounded by the parabolic cylinder {eq}z = 9 - y^2 \; {/eq} and the planes {eq}z = 0, \; x = 3, {/eq} and {eq}x = -3 {/eq}.

## Calculating a Triple Integral using Rectangular Coordinates:

If we are integrating a function {eq}f(x, y) {/eq} over the region bounded by {eq}z = a^2 - y^2, \: -a \leq x \leq a, {/eq} then the triple integral becomes {eq}\displaystyle\iiint f(x, y) \: dV = \int_{-a}^a \int_{-a}^a \int_0^{a^2 - y^2} f(x, y) \: dz \: dy \: dx. {/eq}

The intersection of {eq}z = 9 - y^2 {/eq} with the {eq}xy {/eq} plane is {eq}y = \pm 3. {/eq} The limits of integration are {eq}-3 \leq x \leq 3,...

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