Calculus (differentiation and integration) has a wide application not only for the classical optimization techniques but also for other areas of research including inventory models, nonlinear programming etc.

Today, in real life problems too, Calculus is applied in a major way.** Applications of Differential Calculus**:

Differentiation has many applications, which can be used to find out the rate of change of quantities with respect to time or some other variables, to find the equation of the normal and tangent to a curve at a point, to find out the moving points on the graph of the function which helps to find out the points which has largest or the smallest value throughout the graph, to find out the intervals for which the provided function is increasing or decreasing and of course, to find out the approximate value of certain quantities.

Integration is widely used to find the length of curves, or the area of the region bounded by a curve and a line, or area bounded by two or more curves.

Therefore, f’(x) = 12x

Now, f′(x) = 0 gives x = – 2, 3.

The points x = – 2 and x = 3 divides the real number line into three disjoint intervals, namely, (– infinity, – 2), (– 2, 3) and (3, infinity).

In the intervals (– infinity, – 2) and (3, infinity), f′(x) is positive while in the interval (– 2, 3), f′(x) is negative.

Therefore, the function f is increasing in interval (– infinity, – 2) and (3, infinity) while the function is decreasing in the interval (– 2, 3).

$\frac{dy}{dx}$ = 3x

Now, x y

Thus, required area = $2\int_{0}^{\infty }xdy$ = $2a^{3}\int_{0}^{\infty }$$\frac{1}{a^{2}+y^{2}}$$dy$ = $2a^{3} \times $$\frac{1}{a}$ $\tan^{-1}$$\frac{y}{a}$$|_0^\infty$ = $2a^{2} \times $$\frac{\pi }{2}$ = $\pi a^{2}$.

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