Area bounded by y=f^-1(x) and tangent and normal drawn to it...

Area bounded by $y=f^{-1}(x)$ and tangent and normal drawn to it at the points with abscissae $\pi$ and $2 \pi$, where $f(x)=\sin x-x$ is(ii) $\frac{\pi^{2}}{2}-1$
(b) $\frac{\pi^{2}}{2}-2$
(c) $\frac{\pi^{2}}{2}-4$
(d) $\frac{\pi^{2}}{2}$

Key Concepts

Intersection Points and Limits of Integration

Finding the points where the curve and the lines (such as the tangent and the normal) intersect is critical to accurately define the region whose area is being calculated. Determining these intersection points sets the proper limits of integration, ensuring that the computed area corresponds exactly to the bounded region.

Inverse Functions

Inverse functions switch the roles of the input and output of the original function. In the context of calculus, the derivative of an inverse function is related to the derivative of the original function by the formula (f?¹)'(y) = 1 / f'(x) where y = f(x). This concept is essential when analyzing curves defined by inverses and studying the behavior such as tangents at particular points.

Tangent and Normal Lines

The tangent line to a curve at a given point is the line that just 'touches' the curve at that point, having the same slope as the curve. In contrast, the normal line is perpendicular to the tangent. These lines are crucial in many areas of calculus for approximating curves locally and in applied problems where one needs to determine regions bounded by these linear approximations.

Area Between Curves

Calculating the area between curves involves setting up an integral that represents the region's area by integrating the difference of the functions defining the boundaries. This technique is fundamental in problems where the region is bounded by a curve and one or more lines, requiring careful determination of the limits of integration and the functions that form the upper and lower boundaries.

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