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Application of Integrals

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Multiple Choice Questions

81.

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

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

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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

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

196 Views

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.

640 Views

87.
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

Equation space of space line space AB colon negative space straight y plus 2 space equals fraction numerator 2 plus 3 over denominator 2 end fraction left parenthesis straight x minus 2 right parenthesis rightwards double arrow space 2 straight y space equals space 5 straight x minus 14 Equation space of space line space BC colon negative straight y minus 3 space equals 1 half left parenthesis straight x minus 4 right parenthesis rightwards double arrow space 3 straight y space equals space straight x plus 5 Equation space of space line space CA colon negative left parenthesis straight y minus 2 right parenthesis space equals negative 4 left parenthesis straight x minus 1 right parenthesis 4 straight x plus straight y equals 6 therefore space ar space left parenthesis increment ABC right parenthesis equals integral subscript negative 2 end subscript superscript 3 fraction numerator space 2 straight y space plus 14 over denominator 5 end fraction space dy minus integral subscript 2 superscript 3 3 straight y minus 5 dy space minus integral subscript negative 2 end subscript superscript 2 fraction numerator 6 minus straight y over denominator 4 space end fraction dy equals 75 over 5 minus 5 over 2 minus 24 over 4 equals fraction numerator 300 minus 120 minus 50 over denominator 20 end fraction space equals space 130 over 20 space equals 13 over 2 space sq. space units

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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.

600 Views

89.

Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 2), (1, 5) and (3, 4).

363 Views

90.

Using integration, find the area bounded by the curve x² = 4y and the line x = 4y – 2.

350 Views