Math Assignment Class XII Ch-8 | Applications of Integrations

 Assignment on applications of integral 

Class XII (Mathematics)

Important and extra questions on Applications of Integral for class XII, This assignment is strictly based on previous years CBSE question papers.

Question : 1  Using integration, find the area bounded by the curve 4x² + y² = 36.

Answer: 18π square units

Question : 2 Find the area bounded by the curve, y² = 4x, y-axis, and y = 3.

Answer: 9/4 square units

Question : 3

Find Area bounded by the curve y = x³, the x-axis and the ordinates x = –2 and x = 1 

Answer: 15/4 square units

Solution Hint

Question : 4 

Sketch the graph of y = x|x| and hence find the area bounded by this curve, x – axis and the ordinates x = -2 and x = 2, using integration.

Answer: 16/3 square units

Solution Hint: [ y = x² if x > 0 and y = –x² if x < 0]

 

Question 5: 

Using integration, find the area of the region enclosed between the circle x² + y² = 16 and the lines x = – 2 and x = 2.

Answer: 8√3+16π/3 square units

Question 6: 

Using integration, find the area bounded by the ellipse 9x² +25y² = 225, the line x = - 2, x = 2, and the x-axis.

Answer:   square units

Question 7: 

Using integration find the area of the ellipse   , included between the lines x = -2 and x = 2

Answer: 4√3 + 8π/3 square units

Question 8: 

Find the area of the region bounded by the curves x² = y, y = x + 2 and x-axis, using integration.

Answer: 5/6 square units

Solution Hint:  

Question 9: 

Using integration, find the area of the region bounded by the line y =√3x ,  the curve y =   and y-axis in the first quadrant.

Answer: π/3 square units

Question 10: 

Using integration, find the area of the region bounded by the parabola y² = 4ax and its latus rectum.

Answer: 8/3 a²  square units

Question 11: 

Find the area bounded by the y-axis,  y = cos x and y = sin x when   

Answer: √2 - 1 square units

Solution Hint

Question 12: 

Find the area bounded by the curve y = sin x between x = 0 and x = 2π

Answer: 4 Square units

Solution Hint

  

                                                Or

Question : 13: 

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

Answer: 3/2 Square Units

Question 14: 

Using integration find the area of region bounded by the triangle whose vertices

are (– 1, 0), (1, 3) and (3, 2).

Answer: 4 Square Units

Question 15: 

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

Answer: 8 square units

Question 16: 

If A₁ denotes the area of region bounded by y² = 4x, x = 1 and x-axis in the first quadrant and A₂ denotes the area of region bounded by   y² = 4x, x = 4, find A₁ : A₂.

Answer: A₁  4/3, A₂ = 64/3,  A₁ : A₂ =1:16

Question 17: 

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

Answer: 21/2 square units

Solution Hint: 

Question : 18:  Sketch the graph of y = |x + 3| and evaluate     

Answer: 9 square units

Question : 19: Find the area enclosed by the parabola 4y = 3x² and the line 2y = 3x +12

Answer: 27 Square Unit

Solution Hint:

Question : 20 :  Using the method of integration find the area bounded by the curve |x| + |y| = 1

Answer: 2 Square units

Solution Hint

Question : 21 : Find the area of the smaller region bounded by the ellipse  and the line  .

Answer:    square units

Question : 22 :

Case Study Based Question       (4 marks)

A city park is being redesigned by a landscape architect. The architect wants to create a decorative flower bed enclosed between two curved boundaries. The upper boundary of the flower bed changes depending on the position along the path. The boundaries are represented by the equations: y = x² + 1 and y = x + 1

The flower bed extends from x = 0 to x = 2, and only the region lying above the x-axis is considered for planting.

The architect wants to calculate the exact area of the flower bed to estimate the amount of grass and flowering plants required.

Based on the above information, answer the following questions:

(i) Identify the points of intersection of the curves y = x² + 1 and y = x + 1 in the interval 

0 ≤ x ≤ 2.          (1 Mark)

(ii) Which curve lies above the other in the intervals 0 ≤ x ≤ 1 and 1 ≤ x ≤ 2 ? Justify your answer.           (1 Mark)

(iii) Hence, find the area of the flower bed enclosed by the region    

{(x, y): 0 ≤ y ≤ x² + 1, 0 ≤ y ≤  x + 1, 0 ≤ x ≤ 2.     2 Marks

Answer Key

i) (0,1) and (1,2) 

ii) For 0 ≤ x ≤ 1: x² + 1≤ x + 1  , so the lower curve is y = x² + 1 .

     For 1 ≤ x ≤ 2: x + 1≤ x² + 1  , so the lower curve is y = x +1 . 

iii)  23/6 square units

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