Chapter 8 – Application Of Integrals

Class 12 Mathematics · 37 questions · 0 with answers

Solved examples

example-1Short answer

Find the area of the curve y = sin x between 0 and π. Solution We have Area OAB = = – cos x 0 = cos0 – cosπ = 2 sq units. APPLICATION OF INTEGRALS 171

example-2Short answer

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 2a 2a 1 2 Area BMNC = ∫ xdy = ∫ a y dy 3 3 a a 2a 3a 3 53 = y

example-3Long answer

Find the area of the region bounded by the parabola y2 = 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  1 2 Area = ∫  4 + y – y  dy –2  2  y2 1 3 = 4 y + – y = 18 sq units. 2 6 –2

example-4Short answer

Find the area of the region bounded by the parabolas y2 = 6x and x2 = 6y. 172 MATHEMATICS 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 3 6  x 

example-5Short answer

Find the area enclosed by the curve x = 3 cost, y = 2 sint. Solution Eliminating t as follows: x = 3 cost, y = 2 sint ⇒ = cos t , = sin t , we obtain x2 y 2 + = 1,

example-6Long answer

Find the area of the region included between the parabola y = and the line 3x – 2y + 12 = 0. 3x 2 Solution Solving the equations of the given curves y = and 3x – 2y + 12 = 0, we get 3x2 – 6x – 24 = 0 ⇒ (x – 4) (x + 2) = 0 APPLICATION OF INTEGRALS 173 ⇒ x = 4, x = –2 which give y = 12, y = 3 From Fig.8.6, the required area = area of ABC  12 + 3 x  4 4 2 3x = ∫   dx – ∫ dx –2 2 –2 4 4  3x 2  3x3 = 6 x +  – = 27 sq units.  4  –2 12 −2

example-7Short answer

Find the area of the region bounded by the curves x = at2 and y = 2at between the ordinate coresponding to t = 1 and t = 2. Solution Given that x = at2 ...(i), y = 2at ...(ii) ⇒ t = putting the 2a value of t in (i), we get y2 = 4ax Putting t = 1 and t = 2 in (i), we get x = a, and x = 4a Required area = 2 area of ABCD = 4a 4a 2 ∫ ydx = 2 × 2 ∫ ax dx a a 56 2 = = a sq units.

example-8Long answer

Find the area of the region above the x-axis, included between the parabola y2 = ax and the circle x2 + y2 = 2ax. Solution Solving the given equations of curves, we have x2 + ax = 2ax or x = 0, x = a, which give y = 0. y=±a 174 MATHEMATICS From Fig. 8.8 area ODAB = ∫ ( 2ax – x – ax ) dx Let x = 2a sin2θ. Then dx = 4a sinθ cosθ dθ and x = 0, ⇒ θ = 0, x = a ⇒ θ = . ∫ 2ax – x dx Again, = ∫ ( 2a sin θ cos θ ) ( 4 a sinθ cosθ ) d θ  sin 4θ  4 2 = a2 ∫ (1– cos 4θ ) d θ = a  θ – 2  = 4a . 0  4 0 Further more, 2 3  2 ∫ ax dx = a  x 2  = a2 3  0 3 π 2 2 2  2  Thus the required area = a – a = a2  –  sq units. 4 3 4 3

example-9Long answer

Find the area of a minor segment of the circle x2 + y2 = a2 cut off by the line x = . Solution Solving the equation x2 + y2 = a2 and x = , we obtain their points of a a a 3a  intersection which are  , 3  and  2 , – 2  . 2 2   APPLICATION OF INTEGRALS 175 Hence, from Fig. 8.9, we get Required Area = 2 Area of OAB = 2 ∫a a 2 – x 2 dx x a2 x = 2 2 a2 – x2 + sin –1   2 a a  a2 π a 3 a2 π   . =2 2 2 4– . a – .   2 2 6 `= a2 ( 6π – 3 3 – 2π ) = a2 ( ) 4π – 3 3 sq units.

example-10Multiple choice

The area enclosed by the circle x2 + y2 = 2 is equal to

  • (A)4π sq units
  • (B)2 2π sq units
  • (C)4π sq units
  • (D)2π sq units Solution Correct answer is (D); since Area = 4 ∫ 2 – x x x  =4  2 – x + sin 2 –1  = 2π sq. units. 2 20 x2 y 2
example-11Multiple choice

The area enclosed by the ellipse + = 1 is equal to a 2 b2

  • (A)π2ab
  • (B)πab
  • (C)πa2b
  • (D)πab2 b 2 Solution Correct answer is (B); since Area = 4 ∫ a – x 2 dx 176 MATHEMATICS 4b  x 2 a2 x =  a – x 2 + sin –1  = πab. a 2 2 a 0
example-12Short answer

The area of the region bounded by the curve y = x2 and the line y = 16

example-13Fill in the blanks

The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is _______. Solution sq. units

example-14Fill in the blanks

The area of the region bounded by the curve y = x2 + x, x-axis and the line x = 2 and x = 5 is equal to ________. Solution sq. units

Questions

Q5Short answer

a 5 5 3a 3 = ( 2a ) – a 3 1 55 = a 3 a 3 ( 2)3 – 1 3 2 3 = a 2.2 – 1 sq units.

Q6Short answer

2 3 2 x x Area OABC = ∫  6 x –  dx = 2 6 3 – 18 0 6 2 3 = 2 6 (6) – (6) = 12 sq units. 3 18

Q9Short answer

4 which is the equation of an ellipse. From Fig. 8.5, we get the required area = 4 ∫ 9 – x 2 dx 8 x 9 –1 x  =  9 – x + sin = 6 π sq units. 3 2 2 3  0

This question refers to a figure in the original PDF.

Q32Multiple choice

256 64 128

  • (A)`
  • (B)
  • (C)
  • (D)3 3 3 3 Solution Correct answer is (B); since Area = 2 ∫ ydy Fill in the blanks in each of the Examples 13 and 14.
Q1Short answer

Find the area of the region bounded by the curves y2 = 9x, y = 3x.

Q2Short answer

Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py.

Q3Short answer

Find the area of the region bounded by the curve y = x3 and y = x + 6 and x = 0.

Q4Short answer

Find the area of the region bounded by the curve y2 = 4x, x2 = 4y.

Q5Short answer

Find the area of the region included between y2 = 9x and y = x

Q6Short answer

Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2

Q7Short answer

Find the area of region bounded by the line x = 2 and the parabola y2 = 8x

Q8Short answer

Sketch the region {(x, 0) : y = 4 – x 2 } and x-axis. Find the area of the region using integration.

Q9Short answer

Calcualte the area under the curve y = 2 x included between the lines x = 0 and x = 1.

Q10Short answer

Using integration, find the area of the region bounded by the line 2y = 5x + 7, x- axis and the lines x = 2 and x = 8. APPLICATION OF INTEGRALS 177

Q11Short answer

Draw a rough sketch of the curve y = x –1 in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.

Q12Short answer

Determine the area under the curve y = a 2 – x 2 included between the lines x = 0 and x = a.

Q13Short answer

Find the area of the region bounded by y = x and y = x.

Q14Short answer

Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0.

Q15Short answer

Find the area bounded by the curve y = x , x = 2y + 3 in the first quadrant and x-axis.

Q16Long answer

Find the area of the region bounded by the curve y2 = 2x and x2 + y2 = 4x.

Q17Long answer

Find the area bounded by the curve y = sinx between x = 0 and x = 2π. Find the area of region bounded by the triangle whose vertices are (–1, 1), (0, 5) and (3, 2), using integration.

Q19Long answer

Draw a rough sketch of the region {(x, y) : y2 ≤ 6ax and x2 + y2 ≤ 16a2}. Also find the area of the region sketched using method of integration.

Q20Long answer

Compute the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.

Q21Long answer

Find the area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5.

Q22Long answer

Find the area bounded by the curve y = 2cosx and the x-axis from . x = 0 to x = 2π

Q23Long answer

Draw a rough sketch of the given curve y = 1 + |x +1|, x = –3, x = 3, y = 0 and find the area of the region bounded by them, using integration.

Q24Multiple choice

to 34. π 24. The area of the region bounded by the y-axis, y = cosx and y = sinx, 0 ≤ x ≤ is

  • (A)2 sq units
  • (B)( 2 + 1 ) sq units
  • (C)( 2 – 1 ) sq units
  • (D)( 2 2 – 1 ) sq units
Q25Multiple choice

The area of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is 3 5 7 9

  • (A)sq units
  • (B)sq units
  • (C)sq units
  • (D)sq units 8 8 8 8
Q26Multiple choice

The area of the region bounded by the curve y = 16 − x 2 and x-axis is

  • (A)8 sq units
  • (B)20πsq units
  • (C)16π sq units
  • (D)256π sq units 178 MATHEMATICS
Q27Multiple choice

Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is

  • (A)16π sq units
  • (B)4π sq units
  • (C)32π sq units
  • (D)24 sq units
Q28Multiple choice

Area of the region bounded by the curve y = cosx between x = 0 and x = π is

  • (A)2 sq units
  • (B)4 sq units
  • (C)3 sq units
  • (D)1 sq units
Q29Multiple choice

The area of the region bounded by parabola y2 = x and the straight line 2y = x is 4 2 1

  • (A)sq units
  • (B)1 sq units
  • (C)sq units
  • (D)sq units 3 3 3
Q30Multiple choice

The area of the region bounded by the curve y = sinx between the ordinates π x = 0, x = and the x-axis is

  • (A)2 sq units
  • (B)4 sq units
  • (C)3 sq units
  • (D)1 sq units x2 y 2
Q31Multiple choice

The area of the region bounded by the ellipse + = 1 is 25 16

  • (A)20π sq units
  • (B)20π2 sq units
  • (C)16π2 sq units
  • (D)25 π sq units
Q32Multiple choice

The area of the region bounded by the circle x2 + y2 = 1 is

  • (A)2π sq units
  • (B)π sq units
  • (C)3π sq units
  • (D)4π sq units
Q33Multiple choice

The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is 7 9 11 13

  • (A)sq units
  • (B)sq units
  • (C)sq units
  • (D)sq units 2 2 2 2
Q34Multiple choice

The area of the region bounded by the curve x = 2y + 3 and the y lines. y = 1 and y = –1 is

  • (A)4 sq units
  • (B)sq units
  • (C)6 sq units
  • (D)8 sq units