Solve the given problems by integration. The solar energy E...

Solve the given problems by integration. The solar energy $E$ passing through a hemispherical surface per unit time, per unit area, is $E=2 \pi I \int_{0}^{\pi / 2} \cos \theta \sin \theta d \theta,$ where $I$ is the solar intensity and $\theta$ is the angle at which it is directed (from the perpendicular). Evaluate this integral.

Solve the given problems by integration.
The solar energy $E$ passing through a hemispherical surface per unit time, per unit area, is $E=2 \pi I \int_{0}^{\pi / 2} \cos \theta \sin \theta d \theta,$ where $I$ is the solar intensity and $\theta$ is the angle at which it is directed (from the perpendicular). Evaluate this integral.

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Key Concepts

Definite Integration

Definite integration involves determining the net accumulation of quantities, such as areas under curves, over a defined interval. In mathematical and physical contexts, it is used to calculate exact values by evaluating antiderivatives at the boundaries of the interval, which in applications can represent total energy, mass, charge, etc.

Integration by Substitution

Integration by substitution is a fundamental technique for simplifying integrals by changing variables. This method helps reduce intricate integrals into more basic forms by substituting a part of the integrand with a new variable, thereby transforming the differential and adjusting the integration limits accordingly.

Trigonometric Integrals

Trigonometric integrals are integrals involving sine, cosine, and other trigonometric functions. These types of integrals often arise in applications involving angles and periodic phenomena, and their evaluation can frequently be simplified by using trigonometric identities and substitution techniques.

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