Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Application of Integrals

Name: Zigya - For The Curious Learner
Rating: 4.4 (740 reviews)

Long Answer Type

51. Find the area of the region enclosed between the two circles x² + y² = 4 and (x - 2)² + y² = 4.

892 Views

52. Find the area of the region enclosed between the two circles x² + y² = 1 and (x - 1)² + y² = 1.

565 Views

53. Find the area of the region enclosed between the two circles x² + y² = 1 and

open parentheses straight x minus 1 half close parentheses squared plus straight y squared space equals space 1.

156 Views

Short Answer Type

54. Find the area of the region bounded by two parabola 4 y = x² and 4 x = y²

136 Views

Long Answer Type

55. Find the area included between the two curves y² = 9x and ,x² = 9y. Also draw the rough sketch.

The equations of curves are
$straight y squared space equals 9 straight x$ ...(1)
and    $straight x squared space equals space 9 straight y$ ...(2)
From (1) and (2), we get
   $open parentheses straight x squared over 9 close parentheses squared space equals space 9 space straight x space space space space space space space space or space space space space space space straight x to the power of 4 over 81 space equals space 9 straight x$
$therefore space space space straight x to the power of 4 minus 729 straight x space equals space 0 space space space space space space space space space rightwards double arrow space space straight x left parenthesis straight x cubed minus 729 right parenthesis space equals space 0 therefore space space straight x space equals space 0 comma space space 9 therefore space space straight y space equals space 0 comma space 9$

$therefore$ curves (1) and (2) intersect in O(0, 0), P(9, 9)
From P, draw $PM space perpendicular$ x-axis.
Required area = Area OAPBO = Area OBPM - area OAPM
   $equals space integral subscript 0 superscript 9 square root of 9 straight x end root space dx space minus space integral subscript 0 superscript 9 straight x squared over 9 dx space equals space 3 integral subscript 0 superscript 9 straight x to the power of 1 half end exponent dx minus 1 over 9 integral subscript 0 superscript 9 straight x squared dx$
   $equals space 3 space open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 9 space minus space 1 over 9 open square brackets straight x cubed over 3 close square brackets subscript 0 superscript 9 space equals space 2 space open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 9 space minus space 1 over 27 open square brackets straight x cubed close square brackets subscript 0 superscript 9 equals space 2 space open square brackets 9 to the power of 3 over 2 end exponent minus 0 close square brackets space minus space 1 over 27 open square brackets 9 cubed minus 0 close square brackets equals space 2 space left parenthesis 27 minus 0 right parenthesis space minus space 1 over 27 left parenthesis 729 minus 0 right parenthesis equals space 54 minus 27 space equals space 27 space sq. space units. space$