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Class 10 Class 12

Find the area bounded between the curve y² = x and the line x = 3. Draw the rough sketch also.

The equation of parabola is
y² = x ...(1)
The equation of line is
x = 3
Also. we know that parabola is symmetric about x-axis.
∴ required area = 2 (area ORP)
$equals space 2 integral subscript 0 superscript 3 straight y space dx space equals space 2 space integral subscript 0 superscript 3 space square root of straight x space dx equals space 2 space integral subscript 0 superscript 3 straight x to the power of 1 half end exponent dx space equals space 2 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 3 space equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 3 equals space 4 over 3 open square brackets 3 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times square root of 27 space equals space 4 over 3 cross times 3 square root of 3 space equals space 4 square root of 3 space sq. space units.$

183 Views

Find the area of the region bounded by the curve

straight y space equals space 2 square root of 1 minus straight x squared end root

and x-axis. Draw a rough sketch..

The given equations are

straight y space equals 2 square root of 1 minus straight x squared end root

...(1)
(In the first quadrant)
and y = 0 ...(2)
[∴ x-axis has equation y = 0]
From (1) and (2), we get,

0 space equals space 2 square root of 1 minus straight x squared end root space space space space space space space space rightwards double arrow space space space space space square root of 1 minus straight x squared end root space equals space 0

rightwards double arrow space space space 1 minus straight x squared space equals space 0 space space space space space space space space space space space space rightwards double arrow space space straight x squared space equals space 1 space space space space space space space space rightwards double arrow space space space space straight x space equals space minus 1 comma space space 1

Required area OAB =

integral subscript 0 superscript 1 straight y space dx space equals space 2 integral subscript 0 superscript 1 square root of 1 minus straight x squared space end root space dx

equals space 2 space open square brackets fraction numerator straight x square root of 1 minus straight x squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 equals space open square brackets straight x square root of 1 minus straight x squared end root plus sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 equals left parenthesis 0 plus sin to the power of negative 1 end exponent 1 right parenthesis space minus space left parenthesis 0 plus sin to the power of negative 1 end exponent 0 right parenthesis space equals space 0 plus straight pi over 2 minus left parenthesis 0 plus 0 right parenthesis space equals space straight pi over 2

413 Views

Find the area of the region bounded by the curve y = x² and the line y = 4.

The equation of parabola is
y = x² ...(1)
The equation of line is
y = 4
This parabola is symmetrical about y-axis
$therefore space space space requird space area space POQRP space equals space 2 space left parenthesis area space OQRO right parenthesis$
$equals space 2 space integral subscript 0 superscript 4 straight x space dy space equals space 2 space integral subscript 0 superscript 4 square root of straight y space dy equals space 2 integral subscript 0 superscript 4 straight y to the power of 1 half end exponent dy space equals space 2 open square brackets fraction numerator straight y to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets straight y to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 open square brackets left parenthesis 2 squared right parenthesis to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times 2 cubed space equals space 4 over 3 cross times 8 space equals space 32 over 3 space sq. space units$

220 Views

Give the rough sketch of the curve y² = x and the line x = 4 and find the area between the curve and the line.

The equation of parabola is
y² = x ...(1)
The equation of line is x = 4
Also, we know that parabola is symmetric about x-axis

therefore space space space required space area space equals space 2 space left parenthesis area space ORP right parenthesis

equals space 2 integral subscript 0 superscript 4 space straight y space dx space equals space 2 space integral subscript 0 superscript 4 square root of straight x space dx space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

equals space 2 space integral subscript 0 superscript 4 straight x to the power of 1 half end exponent dx space equals space 2 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 4 space equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 4 space equals space 4 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 0 close square brackets space equals space 4 over 3 cross times square root of 64 space equals space 4 over 3 cross times 8 space equals space 32 over 3 space sq space. units.

150 Views

Calculate the area bounded by the parabola y² = 4 a x and its latus rectum.

The equation of parabola is y² = 4 a x. ...(1)
Let O be the vertex, S be the focus and LL' be the latus rectum of parabola.
The equation of latus rectum is x = a.
Also, we know that parabola is symmetric about x-axis.