Mathematics · Textbook solutions

Application of Integrals

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 14 questions

8.2 Area under Simple Curves

6 q

Solved Examples

Worked · 2
  1. 8.1 Eg.1
    Find the area enclosed by the circle x2+y2=a2x^2 + y^2 = a^2.
  2. 8.1 Eg.2
    Find the area enclosed by the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.

Exercise 8.1

Practice · 4
  1. Ex 8.1 Q1
    Find the area of the region bounded by the ellipse x216+y29=1\frac{x^2}{16} + \frac{y^2}{9} = 1.
  2. Ex 8.1 Q2
    Find the area of the region bounded by the ellipse x24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1.
  3. Ex 8.1 Q3
    Area lying in the first quadrant and bounded by the circle x2+y2=4x^2 + y^2 = 4 and the lines x=0x = 0 and x=2x = 2 is
    1. A.
      π\pi
    2. B.
      π2\frac{\pi}{2}
    3. C.
      π3\frac{\pi}{3}
    4. D.
      π4\frac{\pi}{4}
  4. Ex 8.1 Q4
    Area of the region bounded by the curve y2=4xy^2 = 4x, the yy-axis and the line y=3y = 3 is
    1. A.
      22
    2. B.
      94\frac{9}{4}
    3. C.
      93\frac{9}{3}
    4. D.
      92\frac{9}{2}

Miscellaneous Exercise on Chapter 8

8 q

Solved Examples

Worked · 2
  1. Misc Eg.3
    Find the area of the region bounded by the line y=3x+2y = 3x + 2, the xx-axis and the ordinates x=−1x = -1 and x=1x = 1.
  2. Misc Eg.4
    Find the area bounded by the curve y=cos⁡xy = \cos x between x=0x = 0 and x=2πx = 2\pi.

Miscellaneous Exercise

Practice · 6
  1. Find the area under the given curves and given lines:
    Misc Q1(i)
    y=x2y = x^2, x=1x = 1, x=2x = 2 and xx-axis
  2. Misc Q1(ii)
    y=x4y = x^4, x=1x = 1, x=5x = 5 and xx-axis
  3. Misc Q2
    Sketch the graph of y=∣x+3∣y = |x + 3| and evaluate ∫−60∣x+3∣ dx\int_{-6}^{0} |x + 3|\,dx.
  4. Misc Q3
    Find the area bounded by the curve y=sin⁡xy = \sin x between x=0x = 0 and x=2πx = 2\pi.
  5. Misc Q4
    Area bounded by the curve y=x3y = x^3, the xx-axis and the ordinates x=−2x = -2 and x=1x = 1 is
    1. A.
      −9-9
    2. B.
      −154\frac{-15}{4}
    3. C.
      154\frac{15}{4}
    4. D.
      174\frac{17}{4}
  6. Misc Q5
    The area bounded by the curve y=x∣x∣y = x|x|, the xx-axis and the ordinates x=−1x = -1 and x=1x = 1 is given by [Hint : y=x2y = x^2 if x>0x > 0 and y=−x2y = -x^2 if x<0x < 0].
    1. A.
      00
    2. B.
      13\frac{1}{3}
    3. C.
      23\frac{2}{3}
    4. D.
      43\frac{4}{3}