The area enclosed by the curves y=sinx+cosx and...

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0, \frac{\pi}{2}\right]$ is
[Single Correct Option 2014]
(a) $4(\sqrt{2}-1)$
(b) $2 \sqrt{2}(\sqrt{2}-1)$
(c) $2(\sqrt{2}+1)$
(d) $2 \sqrt{2}(\sqrt{2}+1)$

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Key Concepts

Absolute Value Functions in Integration

When integrating functions that include absolute values, the key concept is to recognize where the expression inside the absolute value is positive or negative. The interval of integration is then divided accordingly, and the expression is replaced by its positive or negative form to compute the integral correctly.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are vital in simplifying complex trigonometric expressions, transforming sums into products, or combining multiple trigonometric functions into a single expression, which is often necessary for easier integration.

Area Between Curves

This concept involves finding the area that is enclosed by two or more curves. The standard approach is to integrate the difference between the top function and the bottom function over the specified interval, effectively summing up the vertical slices of area between the curves.

Definite Integration

Definite integration is used to calculate the exact area under a curve or between curves over a given interval. It involves finding the antiderivative of the function and evaluating it at the endpoints of the interval, which provides a precise measure of the enclosed area.

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