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Application of Integrals

Name: Zigya - For The Curious Learner
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Multiple Choice Questions

81.
Area lying between the curves y² = 4x and y = 2x is

$2 over 3$
$1 third$
$1 fourth$
$1 fourth$

1 third

The equations of the curves are
$straight y squared space equals space 4 straight x$ ...(1)
and    $straight y equals space 2 straight x$ ...(2)
From (1) and (2),
   $4 straight x squared space equals space 4 straight x$
or    $4 straight x left parenthesis straight x minus 1 right parenthesis space equals space 0$ .

$therefore space space space space space space space space space straight x space equals space 0 comma space space 1 therefore space space space from space left parenthesis 2 right parenthesis comma space space straight y space equals space 0 comma space 2 therefore space space space curves space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space intersect space in space straight O left parenthesis 0 comma space 0 right parenthesis comma space space space straight A left parenthesis 6 comma space 2 right parenthesis$

Required area = $integral subscript 0 superscript 1 straight y space dx space equals space integral subscript 0 superscript 1 2 space straight x space dx space equals space 2 space integral subscript 0 superscript 1 straight x to the power of 1 half end exponent dx minus 2 integral subscript 0 superscript 1 straight x space dx space equals space 2 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript 1 space minus space 2 open square brackets straight x squared over 2 close square brackets subscript 0 superscript 1$
$equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript 1 space minus space open square brackets straight x squared close square brackets subscript 0 superscript 1 space equals space 4 over 3 left parenthesis 1 minus 0 right parenthesis minus space left parenthesis 1 minus 0 right parenthesis space equals space 4 over 3 minus 1 space equals space 1 third$

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82. Area bounded by the curve y = x³ the x-axis and the ordinates x - 2 and a = 1 is

-9
$fraction numerator negative 15 over denominator 4 end fraction$
$15 over 4$
$15 over 4$

110 Views

83. The area bounded by the curve y = x |x|, x-axis and the ordinates x = - 1 and x = 1 is given by

0
$1 third$
$2 over 3$
$2 over 3$

122 Views

84. The area of the circle x +y =16 exterior to the parabola y² = 6x is

$4 over 3 left parenthesis 4 straight pi minus square root of 3 right parenthesis$
$4 over 3 left parenthesis 4 straight pi plus square root of 3 right parenthesis$
$4 over 3 left parenthesis 8 straight pi minus square root of 3 right parenthesis$
$4 over 3 left parenthesis 8 straight pi minus square root of 3 right parenthesis$

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85.

The area bounded by the x-axis, y = cosx and y = sin x when $0 less or equal than straight x less or equal than straight pi over 2$

$2 left parenthesis square root of 2 minus 1 right parenthesis$
$square root of 2 minus 1$
$square root of 2 plus 1$
$square root of 2 plus 1$

162 Views

Long Answer Type

86. Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

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87.

Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

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88.

Sketch the region bounded by the curves $straight y equals square root of 5 minus straight x squared end root space and space straight y space equals open vertical bar straight x minus 1 close vertical bar$ and find its area using integration.