Q.The area of the region included between the line $x + y = 1$ and the circle $x^2 + y^2 = 1$ is (A) $\dfrac{\pi}{2} - 1$ (B) $\pi - 2$ (C) $\dfrac{\pi}{4} - \dfrac{1}{2}$ (D) $\pi - \dfrac{1}{2}$

When two curves $y = f(x)$ and $y = g(x)$ cross each other, they enclose a bounded region between them. To find its area, the first step is always to locate the points where the two curves meet, by solving their equations simultaneously -- these intersection points become the limits of integration, and they are almost never given directly in the problem statement.

Once the limits $x = a$ and $x = b$ are known, the enclosed area is the difference between the area under the upper curve and the area under the lower curve, over that same interval, taken as a positive quantity:

In practice this means identifying which of the two curves lies above the other throughout the interval (usually by testing a convenient point between the intersection points), subtracting the lower function from the upper one, and integrating that difference directly -- rather than integrating each curve separately against the X-axis and subtracting afterwards, which is equivalent but usually more work. …