NCERT Solutions for class 12th Mathematics Chapter 8 Application of Integrals

Question 1. Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4, and the x-axis.

Question 2. Find the area of the region bounded by y² = 9x, x = 2, x = 4 and x-axis in the first quadrant.

Question 3. Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Question 4. Find the area of the region bounded by the ellipse x²/16+y²/9 = 1

Note: If a curve makes same area in all 4 quadrants or in 2 quadrants (as in above question). Then we calculate the area in one quadrant and multiply it with 4, if the area is same in 4 quadrants or multiply it with 2, if the area is same in 2 quadrants.

Question 5. Find the area of the region bounded by the ellipse x²/4+y²/9 = 1

Question 6. Find the area of the region in the first quadrant enclosed by x-axis, line

Question 7. Find the area of the smaller part of the circle x² + y² = a² cut off by the line x= a/root2.

Question 8. The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

Question 9. Find the area of the region bounded by the parabola y = x² and y = |x|

Question 10. Find the area bounded by the curve x² = 4y and the line x = 4y – 2

Question 11. Find the area of the region bounded by the curve y² = 4x and the line x = 3

Note: Areas below and above x–axis are equal for a parabola or ellipse or hyperbola have the x-axis as its axis.

Choose the correct answer in the following Questions 12 and 13 :

Exercise 8.2

Question 1. Find the area of the circle 4x² + 4y² = 9 which is interior to the parabola x² = 4y

Note: In case of area between the two curves, we first calculate the points of intersection of the given curves and then integrate with proper limits

Question 2. Find the area bounded by curves (x – 1)² + y² = 1 and x² + y² = 1.

Sol. Given circles are x² + y² = 1

and (x – 1)² + y² = 1 …..(ii)

Centre of …..(i) is O (0, 0) and radius =1

Centre of (ii) is (1, 0) and radius = 1

Both these circle are symmetrical about x-axis

Question 3. Find the area of the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 3.

Question 4. Using integration find the area of region bounded by the triangle whose vertices are (–1, 0), (1, 3) and (3, 2).

Note: If the three vertices of a triangle is given, the area is calculated by first finding the equations of three lines joining these three points and then integrate with proper limits

Question 5. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4

Question 6. Smaller area bounded by the circle x² + y² = 4 and the line x + y =2

Question 7. Area lying between the curves y² = 4x and y = 2x.(a) 2/3 (b) 1/3(c) 1/4(d) 3/4

Miscellaneous Exercise

Question 1. Find the area under the given curves and given lines: (i) y = x², x = 1, x = 2 and x-axis (ii) y = x⁴, x = 1, x = 5 and x-axis

Question 2. Find the area between the curves y = x and y = x².

Question 3. Find the area of the region lying in the first quadrant and bounded by y = 4x², x = 0, y = 1 and y = 4.

Question 4. Sketch the graph of y = |x + 3| and evaluate

Question 5. Find the area bounded by the curve y = sin x between x = 0 and x = 2pi.

Question 6. Find the area enclosed between the parabola y² = 4ax and the line y = mx

Question 7. Find the area enclosed by the parabola 4y = 3x² and the line 2y = 3x + 12.

Question 8. Find the area of the smaller region bounded by the ellipse x²/9 + y²/4 = 1 and the line x/3+ y/2 = 1.

Question 9. Find the area of the smaller region bounded by the ellipse x²/a² + y²/b² = 1 and the line x/a+ y/b = 1

Question 10. Find the area of the region enclosed by the parabola x² = y, the line y = x + 2, and the x-axis

Question 11. Using the method of integration find the area bounded by the curve |x| + |y| = 1.

Question 12. Find the area bounded by curves [(x, y) : y ³ x² and y = |x|}.

Question 13. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A (2, 0), B (4, 5) and C (6, 3)

Question 14. Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0

Question 15. Find the area of the region {(x, y) : y² <= 4x, 4x² + 4y² <= 9}

Choose the correct answer in the following questions from 16 to 19

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