Find the area bounded by the parabola x 2 = 4 y and the straight line x = 4 y - 2. - Wired Faculty

Question

Find the area bounded by the parabola x² = 4 y and the straight line x = 4 y - 2.

Solution

The equation of curve is x² = 4 y    ...(1)
which is upward parabola with vertex O.
The equation of line is
x = 4 y - 2    ...(2)
Let us solve (1) and (2)
Putting x = 4y - 2 in (1), we get
   $left parenthesis 4 space straight y space minus 2 right parenthesis squared space equals space 4 space straight y$
$therefore space space space 16 space straight y squared minus space 16 space straight y plus space 4 space equals space 4 straight y therefore space space 16 space straight y squared minus 20 straight y space plus space 4 space equals space 0 or space 4 straight y squared minus 5 straight y plus 1 space equals space 0 therefore space space space space straight y space equals space fraction numerator 5 plus-or-minus square root of 25 minus 16 end root over denominator 8 end fraction space equals space fraction numerator 5 plus-or-minus 3 over denominator 8 end fraction space equals space 8 over 8 comma space 2 over 8 therefore space space space space straight y space equals space 1 comma space space 1 fourth therefore space space from space left parenthesis 2 right parenthesis comma space space straight x space equals space 4 minus 2 comma space 1 minus 2 space equals space 2 comma space minus 1 therefore space space curve space left parenthesis 1 right parenthesis space and space line space left parenthesis 2 right parenthesis space intersect space in space two space points space straight A left parenthesis 2 comma space 1 right parenthesis space and space straight B space open parentheses negative 1 comma space 1 fourth close parentheses$

From A, draw AM ⊥ x-axis and from B. draw BN ⊥ x-axis.
Required area = area AOB
= Area of trapezium BNMA - (area BNO + area OMA)
= $1 half open parentheses 1 plus 1 fourth close parentheses cross times 3 space minus space integral subscript negative 1 end subscript superscript 2 straight y space dx space equals space 1 half cross times 5 over 4 cross times 3 space minus space integral subscript negative 1 end subscript superscript 2 straight x squared over 4. dx$    $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
$equals space 15 over 8 minus 1 fourth integral subscript negative 1 end subscript superscript 2 straight x squared space dx space equals space 15 over 8 minus 1 fourth open square brackets straight x cubed over 3 close square brackets subscript negative 1 end subscript superscript 2 space equals space 15 over 8 minus fraction numerator 1 over denominator 4 cross times 3 end fraction open square brackets straight x close square brackets subscript negative 1 end subscript superscript 2 equals space 15 over 8 minus 1 over 12 open square brackets left parenthesis 2 right parenthesis cubed minus left parenthesis negative 1 right parenthesis cubed close square brackets space equals space 15 over 8 minus 1 over 12 open square brackets 8 minus left parenthesis negative 1 right parenthesis close square brackets equals space 15 over 8 minus 1 over 12 left parenthesis 8 plus 1 right parenthesis space equals space 15 over 8 minus 9 over 12 space equals space 9 over 8 space sq. space units. space$

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Find the area bounded by the parabola x² = 4 y and the straight line x = 4 y - 2.

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Vector Algebra

Find the area bounded by the parabola x2 = 4 y and the straight line x = 4 y - 2.

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the area bounded by the parabola x² = 4 y and the straight line x = 4 y - 2.