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Mathematics · Textbook solutions

Application of Definite Integration

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 61 questions

  1. 5.1 Solved Ex.1
    Find the area bounded by the curve y=x2y = x^2, the Y axis, the X axis and x=3x = 3.
  2. 5.1 Solved Ex.2
    Find the area of the region bounded by the curves y2=9xy^2 = 9x and x2=9yx^2 = 9y.
  3. 5.1 Solved Ex.3
    Find the area bounded by the curve y=−x2y = -x^2, X-axis and lines x=1x = 1 and x=4x = 4.
  4. 5.1 Solved Ex.4
    Find the area bounded by the line y=xy = x, X axis and the lines x=−1x = -1 and x=4x = 4.
  5. 5.1 Solved Ex.5
    Find the area enclosed between the X-axis and the curve y=sin⁡xy = \sin x for values of xx between 0 to 2π2\pi.
  6. 5.1 Solved Ex.6
    Using integration, find the area of the region bounded by the line 2y+x=82y + x = 8, X-axis and the lines x=2x = 2 and x=4x = 4.
  7. 5.1 Solved Ex.7
    Find the area of the regions bounded by the following curve, the X-axis and the given lines: (i) y=x2y = x^2, x=1x = 1, x=2x = 2 (ii) y2=4xy^2 = 4x, x=1x = 1, x=4x = 4, y≥0y \ge 0 (iii) y=sin⁡xy = \sin x, x=−π2x = -\dfrac{\pi}{2}, x=π2x = \dfrac{\pi}{2}
  8. 5.1 Solved Ex.8
    Find the area of the region bounded by the parabola y2=16xy^2 = 16x and the line x=4x = 4.
  9. 5.1 Solved Ex.9
    Find the area of the region bounded by the curves x2=16yx^2 = 16y, y=1y = 1, y=4y = 4 and the Y-axis, lying in the first quadrant.
  10. 5.1 Solved Ex.10
    Find the area of the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1.
  11. 5.1 Solved Ex.11
    Find the area of the region lying between the parabolas y2=4axy^2 = 4ax and x2=4ayx^2 = 4ay where a>0a > 0.
  12. 5.1 Solved Ex.12
    Find the area of the region bounded by the curve y=x2y = x^2 and the line y=4y = 4.
  13. 5.1 Solved Ex.13
    Find the area of sector bounded by the circle x2+y2=16x^2 + y^2 = 16 and the line y=xy = x in the first quadrant.
  1. Find the area of the region bounded by the following curves, X-axis and the given lines:
    Q.1 (i)
    y=2xy = 2x, x=0x = 0, x=5x = 5
  2. Q.1 (ii)
    x=2yx = 2y, y=0y = 0, y=4y = 4
  3. Q.1 (iii)
    x=0x = 0, x=5x = 5, y=0y = 0, y=4y = 4
  4. Q.1 (iv)
    y=sin⁡xy = \sin x, x=0x = 0, x=π2x = \dfrac{\pi}{2}
  5. Q.1 (v)
    xy=2xy = 2, x=1x = 1, x=4x = 4
  6. Q.1 (vi)
    y2=xy^2 = x, x=0x = 0, x=4x = 4
  7. Q.1 (vii)
    y2=16xy^2 = 16x, x=0x = 0, x=4x = 4
  8. Find the area of the region bounded by the parabola:
    Q.2 (i)
    y2=16xy^2 = 16x and its latus rectum.
  9. Q.2 (ii)
    y=4−x2y = 4 - x^2 and the X-axis
  10. Find the area of the region included between:
    Q.3 (i)
    y2=2xy^2 = 2x, line y=2xy = 2x
  11. Q.3 (ii)
    y2=4xy^2 = 4x, line y=xy = x
  12. Q.3 (iii)
    y=x2y = x^2 and the line y=4xy = 4x
  13. Q.3 (iv)
    y2=4axy^2 = 4ax and the line y=xy = x
  14. Q.3 (v)
    y=x2+3y = x^2 + 3 and the line y=x+3y = x + 3
  1. Misc I (1)
    The area bounded by the region 1≤x≤51 \le x \le 5 and 2≤y≤52 \le y \le 5 is given by
    1. A.
      1212 sq. units
    2. B.
      88 sq. units
    3. C.
      2525 sq. units
    4. D.
      3232 sq. units
  2. Misc I (2)
    The area of the region enclosed by the curve y=1xy = \dfrac{1}{x}, and the lines x=ex = e, x=e2x = e^2 is given by
    1. A.
      11 sq. unit
    2. B.
      12\dfrac{1}{2} sq. unit
    3. C.
      32\dfrac{3}{2} sq. units
    4. D.
      52\dfrac{5}{2} sq. units
  3. Misc I (3)
    The area bounded by the curve y=x3y = x^3, the X-axis and the lines x=−2x = -2 and x=1x = 1 is
    1. A.
      −9-9 sq. units
    2. B.
      −154-\dfrac{15}{4} sq. units
    3. C.
      154\dfrac{15}{4} sq. units
    4. D.
      174\dfrac{17}{4} sq. units
  4. Misc I (4)
    The area enclosed between the parabola y2=4xy^2 = 4x and line y=2xy = 2x is
    1. A.
      23\dfrac{2}{3} sq. units
    2. B.
      13\dfrac{1}{3} sq. units
    3. C.
      14\dfrac{1}{4} sq. units
    4. D.
      34\dfrac{3}{4} sq. units
  5. Misc I (5)
    The area of the region bounded between the line x=4x = 4 and the parabola y2=16xy^2 = 16x is
    1. A.
      1283\dfrac{128}{3} sq. units
    2. B.
      1083\dfrac{108}{3} sq. units
    3. C.
      1183\dfrac{118}{3} sq. units
    4. D.
      2183\dfrac{218}{3} sq. units
  6. Misc I (6)
    The area of the region bounded by y=cos⁡xy = \cos x, Y-axis and the lines x=0x = 0, x=2πx = 2\pi is
    1. A.
      11 sq. unit
    2. B.
      22 sq. units
    3. C.
      33 sq. units
    4. D.
      44 sq. units
  7. Misc I (7)
    The area bounded by the parabola y2=8xy^2 = 8x, the X-axis and the latus rectum is
    1. A.
      313\dfrac{31}{3} sq. units
    2. B.
      323\dfrac{32}{3} sq. units
    3. C.
      3223\dfrac{32\sqrt{2}}{3} sq. units
    4. D.
      163\dfrac{16}{3} sq. units
  8. Misc I (8)
    The area under the curve y=2xy = 2\sqrt{x}, enclosed between the lines x=0x = 0 and x=1x = 1 is
    1. A.
      44 sq. units
    2. B.
      34\dfrac{3}{4} sq. units
    3. C.
      23\dfrac{2}{3} sq. units
    4. D.
      43\dfrac{4}{3} sq. units
  9. Misc I (9)
    The area of the circle x2+y2=25x^2 + y^2 = 25 in first quadrant is
    1. A.
      25π3\dfrac{25\pi}{3} sq. units
    2. B.
      5π5\pi sq. units
    3. C.
      55 sq. units
    4. D.
      33 sq. units
  10. Misc I (10)
    The area of the region bounded by the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 is
    1. A.
      abab sq. units
    2. B.
      πab\pi ab sq. units
    3. C.
      πab\dfrac{\pi}{ab} sq. units
    4. D.
      πa2\pi a^2 sq. units
  11. Misc I (11)
    The area bounded by the parabola y2=xy^2 = x and the line 2y=x2y = x is
    1. A.
      43\dfrac{4}{3} sq. units
    2. B.
      11 sq. units
    3. C.
      23\dfrac{2}{3} sq. units
    4. D.
      13\dfrac{1}{3} sq. units
  12. Misc I (12)
    The area enclosed between the curve y=cos⁡3xy = \cos 3x, 0≤x≤π60 \le x \le \dfrac{\pi}{6} and the X-axis is
    1. A.
      12\dfrac{1}{2} sq. units
    2. B.
      11 sq. units
    3. C.
      23\dfrac{2}{3} sq. units
    4. D.
      13\dfrac{1}{3} sq. units
  13. Misc I (13)
    The area bounded by y=xy = \sqrt{x} and line x=2y+3x = 2y + 3, X-axis in first quadrant is
    1. A.
      232\sqrt{3} sq. units
    2. B.
      99 sq. units
    3. C.
      343\dfrac{34}{3} sq. units
    4. D.
      1818 sq. units
  14. Misc I (14)
    The area bounded by the ellipse x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 and the line xa+yb=1\dfrac{x}{a} + \dfrac{y}{b} = 1 is
    1. A.
      πab−2ab\pi ab - 2ab
    2. B.
      πab4−ab2\dfrac{\pi ab}{4} - \dfrac{ab}{2}
    3. C.
      πab−ab\pi ab - ab
    4. D.
      πab\pi ab
  15. Misc I (15)
    The area bounded by the parabola y=x2y = x^2 and the line y=xy = x is
    1. A.
      12\dfrac{1}{2}
    2. B.
      13\dfrac{1}{3}
    3. C.
      16\dfrac{1}{6}
    4. D.
      112\dfrac{1}{12}
  16. Misc I (16)
    The area enclosed between the two parabolas y2=4xy^2 = 4x and x2=4yx^2 = 4y is
    1. A.
      83\dfrac{8}{3}
    2. B.
      323\dfrac{32}{3}
    3. C.
      163\dfrac{16}{3}
    4. D.
      43\dfrac{4}{3}
  17. Misc I (17)
    The area bounded by the curve y=tan⁡xy = \tan x, X-axis and the line x=π4x = \dfrac{\pi}{4} is
    1. A.
      13log⁡2\dfrac{1}{3}\log 2
    2. B.
      log⁡2\log 2
    3. C.
      2log⁡22\log 2
    4. D.
      3⋅log⁡23\cdot\log 2
  18. Misc I (18)
    The area of the region bounded by x2=16yx^2 = 16y, y=1y = 1, y=4y = 4 and x=0x = 0 in the first quadrant, is
    1. A.
      73\dfrac{7}{3}
    2. B.
      83\dfrac{8}{3}
    3. C.
      643\dfrac{64}{3}
    4. D.
      563\dfrac{56}{3}
  19. Misc I (19)
    The area of the region included between the parabolas y2=4axy^2 = 4ax and x2=4ayx^2 = 4ay, (a>0)(a > 0) is given by
    1. A.
      16a23\dfrac{16a^2}{3}
    2. B.
      8a23\dfrac{8a^2}{3}
    3. C.
      4a23\dfrac{4a^2}{3}
    4. D.
      32a23\dfrac{32a^2}{3}
  20. Misc I (20)
    The area of the region included between the line x+y=1x + y = 1 and the circle x2+y2=1x^2 + y^2 = 1 is
    1. A.
      π2−1\dfrac{\pi}{2} - 1
    2. B.
      π−2\pi - 2
    3. C.
      π4−12\dfrac{\pi}{4} - \dfrac{1}{2}
    4. D.
      π−12\pi - \dfrac{1}{2}
  1. Find the area of the region bounded by the following curve, the X-axis and the given lines
    Misc II Q.1 (i)
    0≤x≤50 \le x \le 5, 0≤y≤20 \le y \le 2
  2. Misc II Q.1 (ii)
    y=sin⁡xy = \sin x, x=0x = 0, x=πx = \pi
  3. Misc II Q.1 (iii)
    y=sin⁡xy = \sin x, x=0x = 0, x=π3x = \dfrac{\pi}{3}
  4. Misc II Q.2
    Find the area of the circle x2+y2=9x^2 + y^2 = 9, using integration.
  5. Misc II Q.3
    Find the area of the ellipse x225+y216=1\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 using integration.
  6. Find the area of the region lying between the parabolas.
    Misc II Q.4 (i)
    y2=4xy^2 = 4x and x2=4yx^2 = 4y
  7. Misc II Q.4 (ii)
    4y2=9x4y^2 = 9x and 3x2=16y3x^2 = 16y
  8. Misc II Q.4 (iii)
    y2=xy^2 = x and x2=yx^2 = y
  9. Misc II Q.5
    Find the area of the region in first quadrant bounded by the circle x2+y2=4x^2 + y^2 = 4 and the X-axis and the line x=y3x = y\sqrt{3}.
  10. Misc II Q.6
    Find the area of the region bounded by the parabola y2=xy^2 = x and the line y=xy = x in the first quadrant.
  11. Misc II Q.7
    Find the area enclosed between the circle x2+y2=1x^2 + y^2 = 1 and the line x+y=1x + y = 1, lying in the first quadrant.
  12. Misc II Q.8
    Find the area of the region bounded by the curve (y−1)2=4(x+1)(y - 1)^2 = 4(x + 1) and the line y=(x−1)y = (x - 1).
  13. Misc II Q.9
    Find the area of the region bounded by the straight line 2y=5x+72y = 5x + 7, X-axis and x=2x = 2, x=5x = 5.
  14. Misc II Q.10
    Find the area of the region bounded by the curve y=4x2y = 4x^2, Y-axis and the lines y=1y = 1, y=4y = 4.