Determination of Areas of Plane Regions

Integral calculus is the branch of mathematics that helps to find the antiderivatives – that are also known as integrals – of a function. It is a subfield of calculus. The process of finding integrals which is the reverse of derivatives is known as integration. 

A family of curves is represented by the integral of a given function. The determination of an area bounded by curves is done through integration or anti-differentiation and is an application of definite integral calculus.

A critical determination of areas of plane regions bounded by curves – applications is to obtain the curve’s equation bordering the area under the graph.

Differential Calculus and Differential Equations

Differential calculus is the branch of mathematics that deals with the study of the rate of change of a quantity with respect to another. It is a subfield of calculus.

The derivative of a function describes the rate of change of a function near a chosen input value. Geometrically, it is the area of the slope of a tangent to the function at that given point in a graph.  

Equations involving derivatives of a dependent variable with respect to an independent variable are known as differential equations. For example:

x (dy/dx) – y = 0 

Differential equations that involve derivatives of a dependent function with respect to more than a single independent variable are known as partial differential equations. For example, (∂2u/∂t2) – c2 (∂2u/∂x2) = 0.

Concept of Integral Calculus

Integral calculus is the branch of mathematics that helps to find the antiderivatives – that are also known as integrals – of a function. It is a subfield of calculus. It deals with the study of integrals and their properties.

The fundamental theorem of calculus relates integral calculus and differential calculus. Integral calculus consists of definite and indefinite integrals. The determination of areas of plane regions bounded by curves – applications belongs to definite integral calculus.

Concepts of Integration and Integrals

Integration is the process of finding integrals and is also known as anti-differentiation. Integrals are the reverse of derivatives. A family of curves is represented by the integral of a given function.

We can generalise integrals based on the type of function and the domain of the performed integration.

Standard integration formulas where, C is an arbitrary constant are:

• ∫ dx = x + C

• ∫a dx = ax + C

• ∫(1/x) dx = ln |x| + C

• ∫eX dx = eX + C

• ∫ax dx = (ax / ln a) + C

• ∫ln x dx = x ln x – x + C

• ∫sin x dx = -cos x + C

• ∫cos x dx = sin x + C

• ∫tan x dx = ln |sec x| + C or C – ln |cos x|

• ∫cot x dx = ln |sin x| + C

• ∫sec x dx = ln |sec x + tan x| + C

• ∫cosec x dx = ln |cosec x – cot x| + C

• ∫sec2 x dx = tan x + C

• ∫sec x tan x dx = sec x + C

• ∫cosec2x dx = -cot x + C

• ∫tan2 x dx = tan x – x + C

Bounded Area between Two Curves and its Formula of Expression

The area between two curves is the area of the graph’s region bounded by the intersecting curves. We can calculate this area using integral calculus.

To find the area between the two curves, first, we divide the bounded area into several small rectangular strips parallel to the y-axis, starting from x = a to x = b. We can add these small areas to obtain the area under the curve by using integration.

The width and height of these rectangular strips is dx and f(x) – g(x), respectively. Hence, the area is given by:

[f(x) – g(x)] dx

Now, we can obtain the area bounded by the two curves by integrating the above area within limits x = a to x = b. By assuming, f(x) and g(x) are continuous on [a, b] and g(x) < f(x) ∀ x ∈ [a, b], we get the formula for bounded area between two curves.

The formula is expressed as:

Area of plane regions bounded by curves = ∫ab [f(x)−g(x)] dx

A point to be remembered is that the areas of plane regions bounded by curves are always a non-negative value. It can be 0 in decimals or fractions, but never negative.

Example: Find the area between two curves f(x) = x2 and g(x) = x3 within the interval [0,1]

Solution:f(x)=x2 and g(x)=x3

In the region [0,1] f(x)>=g(x) 

So area between the curves will be- 

∫1(x2-x3)=[x3/3-x4/4]10=1/12

0

Determination of Areas of Plane Regions Bounded by Curves – Applications

One of the important determinations of areas of plane regions bounded by curves – applications is to obtain the equation of the curve bounding the area under the graph.

The area under the curves represents the bounded area enclosed under the curves in a Cartesian plane marked with limiting points. It gives the area of the irregular plane shape in a two-dimensional array.

Conclusion

This easy-to-understand and well-written article on Determination of Areas of Plane Regions Bounded by Curves – Applications through integration and differential equations. It familiarises one with the concepts of integral calculus and integration, standard formulas for integral calculus, the bounded area between two curves and its formula of expression, and the determination of areas of plane regions bounded by curves – applications.

We now know that the determination of areas of plane regions bounded by curves – applications belong to definite integral calculus. We also know that it is among the crucial applications to obtain the equation of the curve bounding the area under the graph.