We have an answer to "Evaluate the integral. ∫01∫0π∫0πy⋅cos⁡(z)dxdydz"

Brennan Flores

Answered question

2021-10-25

Evaluate the integral.

\int_{0}^{1} \int_{0}^{π} \int_{0}^{π} y \cdot \cos (z) d x d y d z

Demi-Leigh Barrera

Skilled2021-10-26Added 97 answers

Step 1
Given that:

\int_{0}^{1} \int_{0}^{π} \int_{0}^{π} y \cdot \cos (z) d x d y d z

Step 2
As the given integral have constant limits.
So, each integral can be evaluated separately.
Thus,

\int_{0}^{1} \int_{0}^{π} \int_{0}^{π} y \cdot \cos (z) d x d y d z = \int_{0}^{1} 1 d x \int_{0}^{π} y d y \int_{0}^{π} \cos z d z

= {[x]}_{0}^{1} \cdot {[\frac{y^{2}}{2}]}_{0}^{π} {[\sin z]}_{0}^{π}

= [1 - 0] \cdot [\frac{π^{2} - 0^{2}}{2}] [\sin π - \sin 0]

= 1 \times \frac{π^{2}}{2} \times 0

=0
So,

\int_{0}^{1} \int_{0}^{π} \int_{0}^{π} y \cdot \cos (z) d x d y d z = 0

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