(4) $$ \int_{0}^{\pi / 2} \int_{0}^{\cos x} y \sin x d y d z $$

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Calculate the integral $$ \int_{0}^{\pi / 2} \int_{0}^{\cos x} y \sin x d y d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{0}^{\frac{\pi }{2}} \int_{0}^{\cos\left(xight)} y\sin\left(xight) dy dx$$
- step1: Evaluate the inner integral:
  $$\int_{0}^{\frac{\pi }{2}} \frac{1}{2}\cos^{2}\left(xight)\sin\left(xight) dx$$
- step2: Evaluate the integral:
  $$\int \frac{1}{2}\cos^{2}\left(xight)\sin\left(xight) dx$$
- step3: Use properties of integrals:
  $$\frac{1}{2}	imes \int \cos^{2}\left(xight)\sin\left(xight) dx$$
- step4: Evaluate the integral:
  $$\frac{1}{2}\left(-\frac{\cos^{2+1}\left(xight)}{2+1}ight)$$
- step5: Simplify:
  $$\frac{1}{2}\left(-\frac{1}{3}\cos^{3}\left(xight)ight)$$
- step6: Rewrite the expression:
  $$-\frac{1}{2}	imes \frac{1}{3}\cos^{3}\left(xight)$$
- step7: Multiply the terms:
  $$-\frac{1}{6}\cos^{3}\left(xight)$$
- step8: Return the limits:
  $$\left(-\frac{1}{6}\cos^{3}\left(xight)ight)\bigg |_{0}^{\frac{\pi }{2}}$$
- step9: Calculate the value:
  $$\frac{1}{6}$$
The value of the given double integral $$ \int_{0}^{\pi / 2} \int_{0}^{\cos x} y \sin x d y d x $$ is $$ \frac{1}{6} $$ or approximately 0.1667.
The value of the integral is $$ \frac{1}{6} $$ or approximately 0.1667.

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