CBSE Class 12 Mathematics Notes Chapter 8 Application of Integrals

The application of integrals is used to calculate the area and volume of different 2-D and 3-D curves, and they have various applications in mathematics. 

• They help us calculate the area of an arc of a circle, irregular boundaries, the volume of various curves, and the area between the two curves. 
• Integrals include the summation of discrete data, and their applications cover basic integral concepts such as the fundamental theorem of calculus. 
• It is used in the fields of architecture, electrical engineering, medical science and statistics.
• In mathematics, integrals are used to calculate the area under a curve and the area of a region bounded by a curve and a line.

Application of integrals is important for class 12 students, as well as engineering mathematics. CBSE Class 12 Mathematics Notes for Chapter 8 Application of Integrals are given in the article below for easy preparation and understanding of the concepts involved.

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Application of Integrals

There are various applications of integrals. Some of them are as follows:

• It is used to calculate the area under simple curves.
• The concept is used to find areas enclosed by lines, arcs of circles, parabolas, and ellipses.
• Integrals are used to find out the area between two curves.
• It is used to find out the centroids of areas of the triangle with curved boundaries.
• In the field of statistics, it is used to determine survey data to improve marketing plans for different companies.

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Area Between the Curve and the Axis

• The area of the region bounded by the curve y = f(x), x-axis, and the lines x = a and x = b (b > a) is given by the formula

Area = ∫_a^b y dx = ∫_a^b f(x) dx

The area between the curve and the axis

• The area of the region bounded by the curve x = Φ(y), y-axis, and the lines y = c and y = d (b > a) is given by the formula

Area = ∫_c^d x dy = ∫_c^d Φ(y) dy

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Area Between Two Curve

• The area of the region enclosed between two curves y = f(x), y = g(x), and the lines x = a, x = b is given by the formula

Area = ∫_a^b [f(x) - g(x)] dx

Where f(x) ≥ g (x) in [a, b]

Area Between Two Curve

• If f(x) ≥ g (x) in [a, c] and f(x) ≤ g (x) in [c, b], a < c < b, then

Area = ∫_a^c [f(x) - g(x)] dx + ∫_c^b [g(x) - f(x)] dx

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Method to Find Area Under Curve

• To calculate area, first, find the equation of the curve, y = f(x), as well as its limits and axis.
• The integration i.e. antiderivative of the curve is found.
• The upper and lower limits are applied to the integral result, and the difference provides the area under the curve.

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Area in Polar Coordinates

• Consider the region OKM bounded by a polar curve r = f(θ) and two semi-straight lines θ = ⍺ and θ = ꞵ.
• The area of the polar region is given by

Area = 1/2 ∫_⍺^ꞵ r² dθ = 1/2 ∫_⍺^ꞵ f²(θ) dθ

Area in Polar Coordinates

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Area Between Two Polar Curves

• The area of a region between two polar curves r = f(θ) and r = g(θ) in the sector [⍺, ꞵ] is expressed by the integral

Area = 1/2 ∫_⍺^ꞵ [f²(θ) – g²(θ)] dθ

Area Between Two Polar Curves

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Some Standard Curves and their Equation

• Straight Line:

x = a and x = – a, where a > 0

Straight Line

• Circle:

x² + y² = a²

Circle

• Parabola:

y² = 4ax or y² = – 4ax

Parabola

• Ellipse

x²/a² + y²/b² = 1

Ellipse

There are Some important List Of Top Mathematics Questions On Applications Of Integrals Asked In CBSE CLASS XII