CBSE Class 12 Maths Revision Notes Chapter 8

NCERT Class 12  Mathematics Chapter 8 Notes

Mathematics is an essential subject that forms the basis of all other subjects. It encourages logical reasoning and is necessary for business, finance and personal decision making. In this subject, strong basic knowledge is  required to understand complex subjects and solve problems based on them.. The majority of students face issues and find it difficult to tackle problems in mathematics. To easily understand the concepts, methods and gain knowledge of the higher-level topics, students must  learn from Class 12 Mathematics Chapter 8 notes and get an understanding of all the basic concepts. 

The Class 12 Chapter 8 Mathematics notes- Application of Integrals is a continuation of Chapter 7. Integration is an important concept  in higher mathematics. This Chapter deals with the calculation of complex areas under curves, parabolas, ellipses, and intercepts. Students also learn about various integration techniques and formulas. With the help of the Class 12 Mathematics Chapter 8 notes, students can strengthen their foundation and solve problems based on the real world.  

Extramarks make Mathematics interesting and fun with coloured illustrations and detailed information. The Class 12 Mathematics Chapter 8 notes help students to practice unlimited questions and attain high scores in the examination. Students can also access different practice tests, mock tests, reference books, etc.,  from the Extramarks web portal. 

NCERT Class 12 Mathematics Chapter 8: Key Notes

The main topics covered in the Class 12 Mathematics Chapter 8 notes are:

• Introduction
• Definite integrals
• Newton Leibnitz’s Theorem 
• Area under Simple Curves
• Area between Two Curves.

Introduction: 

The Chapter begins by recalling the concepts of finding areas bounded by the curve and definite integrals. It also introduces the various applications of integrals in this Chapter and topics like areas under simple curves. Between lines,  and curves, parabola, and ellipses are also taught. Calculation of the average value of a function with the help of integration is included in the Chapter. 

Real-life problems such as a record of rainfall in a day expressed in the form of a curve with specified limit x to limit y are included in the Class 12 Mathematics Chapter 8 notes. 

Definite Integrals:

Let F(x) be the antiderivative of function f(x), then the definite integral of f(x) from a to b is given as F(b) – F(a), such that variable x has any two independent values a and b. It is denoted as abf(x) dx.

Therefore, we can say that  abf(x) dx= F(b) – F(a). the values a and b are called the limits of integration. 

Properties: 

1. abf(x) dx = –baf(x) dx
2. abf(x) dx = baf(y) dy
3. abf(x) dx = acf(x) dx + cbf(x) dx, where a< c <b
4. 0af(x) dx = abf(a-x) dx
5. abf(x) dx =abf(a+b-x) dx
6. 0af(x)f(x)+f(a-x) dx = a2
7. abf(x)f(x)+f(a+b-x) dx = b-a2
8. 02af(x) dx = 0af(x) dx + 0af(2a-x) dx 

If f(x) is a periodic function, i.e. f(a+x) = f(x) then,

1.  0naf(x) dx=  n0af(x) dx
2.  0naf(x) dx= (n-1)  0af(x) dx
3.  0b+naf(x) dx=  0bf(x) dx
4. If f(x) 0 on [a, b], then  abf(x) dx 0
5. If f(x) g(x) on [a, b], then   abf(x) dx   abg(x) dx
6. abf(x) dx abf(x) dx 
7. abf(x) dx = f(c) (b – a), for a < c < b

Newton Leibnitz’s Theorem:

Consider two differentiable functions g(x) and h(x) for x [a, b], the function f is continuous in interval [a, b] then 

ddxg(x)h(x)f(x) dx =ddxh(x) . f(h(x)) – ddxg(x) . f(g(x))

Definite Integral as a Limit of Sum:

The function f(x) is continuous on the interval [a, b] divided into n parts, then

abf(x) dx= nr=0n=1(b-a)n f(a+ (b-a) rn

Reduction Formulae in Definite Integrals

1. If In= o2sinnx dx then In=(n-1n) In-2

NOTE: In= o2sinnx dx  = o2cosnx dx

1. If In= o4tannx dx then In+ In-2= 1n-1
2. If In= o2sinmx. cosnx dx then Im,n=(m-1m+n) Im-2, n

The area under the curves:

1. The total area A of the region bounded between the x-axis with co-ordinates x = a, x = b and the curve y = f (x)  is given as abdA = aby dx =abf(x) dx.

If f(x)>0, ∀x∈[a,c) and f(x)<0 ∀x∈(c,b], then

Area = acf(x) dx+ cbf(x) dx = acf(x) dx – cbf(x) dx  ∀ a < c < b

        2. The area of the given region bounded between the curve x = g (y), y-axis and the lines y = c,

y = d is given as cddA = cdx dy =cdg(y) dx.

NOTE: If the curve is below the x-axis, then f (x) < 0 from x = a to x = b. In this case, the area will be negative. But since only the numerical value is taken into consideration, we take the absolute value of the area, which is given by abf(x) dx

The area of the region bounded by a curve and a line:

The area of the region is bounded by a line and a circle, parabola, or an ellipse in their standard forms. Vertical stripes or horizontal stripes are used to calculate the area of the region. 

For, e.g. Consider the figure given below. We have to find the area bounded by the ellipse in its standard form and the ordinates x = 0 and x = ae, where b2= a2(1- e2) and e < 1. The area of the highlighted region is enclosed by the lines x = 0 and x = ae and the eclipse. Therefore using the formula for area, we will integrate and find the solution.

The area of the region between two curves:

Here total area = ab[f(x)-g(x)]dx i.e.,

Area A= [area bounded by the curve y = f (x), x-axis and lines x = a, x = b] – [area bounded by the curve 

y = g (x), x-axis and lines x = a, x = b]

A = abf(x)dx –abg(x)dx =  ab[f(x)-g(x)]dx where f(x) > g(x) in the interval [a, b]

In this case, the total area A= Area of the ACBDA region+ Area of the BPRQB region.

Area = ac[f(x)-g(x)]dx + cb[g(x)-f(x)]dx, where a < c < b and f (x) ≥ g (x) in interval [a, c] and f (x) ≤ g (x) in the interval [c, b]. 

Curve Tracing:

To locate the area of a region, it is necessary to draw a rough sketch. Consider the curve f(x,y) = 0. To find the area of the curve, follow the steps given below:

Step 1: Symmetry 

• The curve is said to be symmetric about the x-axis if all the powers of y in the equation are even. 
• The curve will be symmetric about the y-axis if all the powers of x in the equation are even. 
• The curve is symmetric about line y = x if the given equation remains unchanged on interchanging the value of x and y. 
• The curve will be symmetrical in opposite quadrants if the given equation remains unchanged when x and y values are replaced by -x and -y. 

Step 2: Origin

• If the constant term is absent in the given equation, then we can say that the curve passes through the origin (0,0). 
• Then calculate the tangents at the point (0, 0) by equating the terms having the lowest degree in the given equation to zero. 

Step 3: Intersection with Co-ordinates Axes

• Find values of x by substituting y=0 to estimate the intersecting points of the curve with an x-axis
• Find values of y by substituting x=0 to estimate the intersecting points of the curve with the y-axis

Step 4: Asymptotes

• Compare the coefficient of the highest power of variable y in the given algebraic equation to zero to find out the vertical asymptotes. 
• Compare the coefficient of the highest power of variable x in the given algebraic equation to zero to find out the horizontal asymptotes. 

Step 5: Region

• Solve the given algebraic equation for x in terms of y or vice versa to determine the regions in which the curve doesn’t lie. 

Step 6: Critical Points

• Differentiate the value of y with respect to x and find out which values of x satisfy ddxy = 0 

Step 7: Trace the given curve

Chapter 8 Mathematics Class 12 Notes: Exercises & Answer Solutions

The Class 12 Mathematics Chapter 8 notes ensure detailed and apt information of all the concepts for students to get a clear understanding. Students learn to calculate the area of different regions bounded by curves, lines, parabolas and ellipses. With the help of the Class 12 Mathematics Chapter 8 notes, students can get all important definitions, formulas, properties, and theorems in one place to enable quick revision to clear their doubts and provide them with a solid foundation.. 

Click on the links given below to gain access to the Extramarks Questions & Answers  of this chapter.

Extramarks, an online learning platform, aims to provide a fun and engaging learning experience. The  Class 12 Mathematics Chapter 8 notes are prepared by academic experts by analysing various CBSE sample papers and CBSE previous year question papers. Extramarks provides various study material and CBSE revision notes to help students in the exam preparation. 

NCERT Exemplar Class 12 Mathematics

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The NCERT Exemplar is based on the latest CBSE syllabus. Students must regularly practice all the important questions to attain high scores in the examination. It also teaches several shortcut techniques to tackle complex problems easily in no time. With the help of academic notes such as NCERT Exemplar and Extramarks Class 12 Mathematics Chapter 8 notes, students can study hassle-free  according to  CBSE pattern. 

NCERT Class 12 Mathematics Chapter 8 Notes: Key Features

The key features of Extramarks  Class 12 Mathematics Chapter 8 notes are as under.

• The notes provide all concepts included in the CBSE syllabus in a detailed and lucid manner.
• It is prepared by subject matter experts at Extramarks
• Students get an idea of the marking system, weightage and  exam pattern of the CBSE examinations. 
• The Class 12 mathematics notes Chapter 8 provides authentic knowledge and helps to clarify all doubts or queries way ahead of the exam to put the students at ease. 
•  Inadvertently students  develop time management and problem-solving skills.