Applications Of Integrals For Class 12

There are numerous ways to calculate, including the use of functions, differentiation, and integration. Numerous disciplines, including mathematics, science, and engineering, use integrals. The majority of the formulas we use to calculate areas are integral formulas. Integrals are used to compute areas under curves, regions bounded by any curve, as well as other areas in mathematics. The main applications of integration are computing the volumes of three-dimensional objects and determining the areas of two-dimensional regions. The use of integrations in practical situations depends on the types of industries in which this calculus is applied.

What Is An Integral?

Being the opposite of differentiation, the integral is also known as the anti-derivative. Integration describes the procedure for obtaining the antiderivative. The term "integral" refers to the value of the function determined during the integration process. Calculating the area to the X-axis from the curve is what is implied by finding the integral of a function with respect to x.

Types Of Integrals

The two main categories of integrals are:

• Definite Integrals- The term "definite integral" refers to an integral that has upper and lower limits, thereby making the integral's final value definite. The area under a curve with respect to one of the coordinate axes and within the specified bounds is determined using the definite integrals. Another name for it is the "Riemann Integral”. It is portrayed as:

\int_{a}^{b}g(x)dx=G(b)-G(a)

• Indefinite Integrals- The definition of an indefinite integral is an integral whose upper and lower boundaries are unknown, consequently, the integral's final value is indefinite. The exponential, logarithmic, and trigonometric functions, as well as algebraic expressions, are all integrated using indefinite integrals.A constant is therefore added to the integral's result because the integration does not return the original expression's constant value. It is shown as below:

\int g(x)dx=G(x)+constant

Applications Of Integrals

Integrals find their applications in various fields, some of these are mentioned below:

In the field of maths

• To determine the centroid (center of mass) of a region with curved sides.

• To calculate the area between two curves.

• To identify the area beneath a curve.

In the field of physics

• To calculate the center of gravity.

• To find the center of mass.

• To calculate the mass and momentum of inertia.

In daily life

• To compute the growth of bacteria in medical science.

• To evaluate survey data in statistics.

How To Find The Area Under A Curve?

Three easy steps can be used to compute the area under the curve.

Step 1: The curve's equation, the boundaries across which the area is to be determined, and the axis containing the area must all be known.

Step 2: The integration (antiderivative) of the curve must be determined.

Step 3: Finally, in order to determine the area under the curve, we must apply the upper limit and lower limit to the integral result and subtract them.

For the curve y=f(x), the area under the curve is calculated as follows:

=\int_{a}^{b}ydx

=\int_{a}^{b}f(x)dx

=[g(x)]_a^b

=g(b)-g(a)

Points To Remember

• An integral is the value of the function determined through the integration process.

• Definite Integrals and Indefinite Integrals are the two main types of integrals.