NCERT Exemplar Class 12 Maths Unit 8 Application of Integrals

Candidates can download NCERT Exemplar Class 12 Maths Unit 8 from this page. The exemplar has been provided by the National Council of Educational Research & Training (NCERT) and the candidates can check it from below for free of cost. It contains objective, very short answer type, short answer type, and long answer type questions. Along with it, the answer for each question has also been provided. From the NCERT Exemplar Class 12 Maths Unit 8, candidates can understand the level and type of questions that are asked in the exam.

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NCERT Exemplar Class 12 Maths Unit 8 Application of Integrals

NCERT Class 12 Maths Unit 8 is for Application of Integrals. The type of questions that will be asked from NCERT Class 12 Maths Unit 8 are displayed in the below provided NCERT Exemplar Class 12 Maths Unit 8. With the help of it, candidates can prepare well for the examination.

Also Check: NCERT Solutions for Class 12 Maths

Overview
This chapter deals with a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses, and finding the area bounded by the above said curves.

1. 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:

2. The area of the region bounded by the curve x = Φ (y), y-axis and the lines y = c, y = d is given by the formula:

3. 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.

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

Short Answer Type Questions (Solved Examples)

Example 1 Find the area of the curve y = sin x between 0 and π.
Solution We have

Example 2 Find the area of the region bounded by the curve ay² = x³ , the y-axis and
the lines y = a and y = 2a.
Solution We have

Example 3 Find the area of the region bounded by the parabola y 2 = 2x and the straight line x – y = 4.
Solution The intersecting points of the given curves are obtained by solving the equations x – y = 4 and y² = 2x for x and y. We have y² = 8 + 2y i.e., (y – 4) (y + 2) = 0 which gives y = 4, –2 and x = 8, 2.
Thus, the points of intersection are (8, 4), (2, –2). Hence

Example 4 Find the area of the region bounded by the parabolas y² = 6x and x² = 6y.
Solution The intersecting points of the given parabolas are obtained by solving these equations for x and y, which are 0(0, 0) and (6, 6). Hence

Example 5 Find the area enclosed by the curve x = 3 cost, y = 2 sint.
Solution Eliminating t as follows:

Long Answer Type Questions (Solved Examples)

Example 7 Find the area of the region bounded by the curves x = at² and y = 2 at between the ordinate
coresponding to t = 1 and t = 2.
Solution Given that x = at² …(i),