"Evaluate the integrals by changing the order of..." was answered by our experts

Asher Kaufman

Open question

2022-08-18

Evaluate the integrals by changing the order of integration in an appropriate way.

\int_{0}^{4} \int_{0}^{1} \int_{2 y}^{2} \frac{4 \cos (x^{2})}{2 \sqrt{z}} d x d y d z

ellynnauwh

Beginner2022-08-19Added 7 answers

Change the order of integration from dxdydz to dydxdz.
Now we have:

\int_{0}^{4} \int_{0}^{1} \int_{2 y}^{2} \frac{4 \cos (x^{2})}{2 \sqrt{z}} d x d y d z = \int_{0}^{4} \int_{0}^{2} \int_{0}^{\frac{1}{2} x} \frac{4 {\cos x}^{2}}{2 \sqrt{z}} d y d x d z

= \int_{0}^{4} \int_{0}^{2} \frac{4 {\cos x}^{2}}{2 \sqrt{z}} \frac{x}{2} d x d z = \int_{0}^{4} \int_{0}^{2} \frac{x {\cos x}^{2}}{\sqrt{z}} d x d z

= \int_{0}^{4} {[\frac{{\sin x}^{2}}{2 \sqrt{z}}]}_{x = 0}^{x = 2} d z

= \sin 4 {[\sqrt{z}]}_{z = 0}^{z = 4}

= 2 \sin 4

Result:

2 \sin 4

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