Solved Problems on Definite Integral

The area of the curve in a graph could be calculated using the Definite Integral. It has start and endpoints wherein the area under a curve is determined and has limits. To determine the area of the curve f(x) with respect to the x-axis, use the limit points [a, b]. The analogous definite integral expression is abf(x)dx. The sum of the areas is integration, and definite integrals are utilised to find the area within limitations.

Integration was first studied in the third century BC when it was used to calculate the area of circles, ellipses and parabolas. Let’s get a better understanding of definite integrals and their properties.

Define Definite Integral?

The two most important notions in calculus are differentiation and integration. Calculus is a discipline of mathematics concerned with the study of issues in which the values of quantities change continuously. Differentiation is the process of breaking down a complex function into smaller components. Integration is the process of combining smaller functions to generate a larger unit. Integration and differentiation are inverse mathematical operations because they are opposed. Integrals are divided into two categories. There are two types of integrals: definite and indefinite.

The area under a curve between two set bounds is called a definite integral. For a function f(x) defined with reference to the x-axis, the definite integral is written as abf(x)dx, where a is the lower limit and b is the upper limit. We split the area under a curve between two limits into rectangles and added them to find the area under the curve. The more rectangles there are, the more precise the region is. As a result, we divide the area into an unlimited number of rectangles, each of which is the same (very small) size, and then add all of the areas together.

The value of the definite integral for the higher limit minus the value of the definite integral for the lower limit gives the final value of a definite integral.

Solved Problems on  Definite Integral Meaning and Formulas 

The definite integral qualities aid in calculating the integral for a function multiplied by a constant, the sum of functions, and even and odd functions. Let’s examine some of the features of definite integrals that can help us solve problems with definite integrals.

•  abf(x)dx=abf(t)dt
• abcf(x)dx=cabf(x)dx
• abf(x)dx=−baf(x)dx
• abf(x)dx=acf(x).dx+ cbf(x).dx
• abf(x)dx=abf(a+b−x).dx
• ab(f(x)±g(x).)dx=abf(x).dx±abg(x).dx
• a0f(x).dx=a0f(a−x).dx
• 02af(x).dx=20af(x).dx, if f(2a – x) = f(x)
• -aaf(x).dx=20af(x).dx, if f(x) is an even function (i.e., f(-x) = f(x)).
• 02af(x).dx=0, if f(2a – x) = -f(x).
• -aaf(x).dx=0, if f(x) is an odd function (i.e., f(-x) = -f(x)).

Solved Problems on  Definite Integral Examples and Applications 

Definite integrals are the concept that is used in different Mathematical operations such as:-

• Calculating the area between two cubic curves, quadratic and linear
• Determining the volumes
• Calculating the length of the plane curve
• Determining the surface area of the revolution

Apart from these calculations, definite integrals are used in various other domains, including physics, statistics and engineering.

Determining the Area of the Circle by Definite Integral

The area of a circle is determined by first determining the area of the first quadrant of the circle. The circle’s equation, x2 + y2 = a2, is converted to a curve’s equation,y = √(a2 – x2). To get the solution of the curve with reference to the x-axis & the limits from 0 to a, we apply the idea of the definite integral. The area of the circle is four times that of the circle’s quarter. The area of the quadrant is derived by integrating the curve’s equation across the first quadrant’s bounds.

A = 40ay.dx

= 40a√a2−x2.dx

= 4[x2√a2−x2+a22sin-1(x/a)]a0

= 4[((a/2)× 0 + (a2/2)sin-11) – 0]

= 4(a2/2)(π/2)

= πa2

Determining the Area of the Parabola by Definite Integral

The axis of a parabola divides the parabola into two symmetric sections. We’ll use a parabola with the equation y2 = 4ax symmetric along the x-axis. Using the definite integral formulas with respect to the x-axis and the limits from 0 to a, we first find the area of the parabola in the first quadrant. The definite integral is calculated and then doubled within the boundary to get the area under the entire parabola.

Therefore the area under the curve enclosed by the parabola is 8a2/3 square units.

Determining the Area of the Ellipse by Definite Integral

x2/a2 + y2/b2 = 1 is the equation for an ellipse with a major length 2a and a minor axis of length 2b. This equation can be expressed as y = bx/a (a2 – x2). The area circumscribed by the ellipse in the first coordinate and with respect to the x-axis is calculated using the idea of a definite integral. It is then multiplied by four to produce the ellipse’s area. On the x-axis, the boundary limitations range from 0 to a. Therefore, the area of the ellipse is πab sq units

Conclusion

We have read about the Definite Integral and Important Properties problems. In mathematics, an integral assigns numbers to equations to express displacement, volume, area, and other concepts arising from connecting infinitesimal data.The term “integration” refers to the process of locating integrals. When the limitations must be defined to obtain a unique value, defined integrals are utilised. When the integrand’s bounds aren’t stated, indefinite integrals are used. If the lower and upper limits of a function’s independent variable are known, the integration is described using definite integrals.