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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

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

196 Views

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

-9

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

15 over 4

We want to find the area bounded by the curve y = x³, the x-axis and the ordinates x = - 2 and x = 1.
A rough sketch of the curve y = x³ is given in the figure.

Required area = $integral subscript negative 2 end subscript superscript 1 space open vertical bar straight x cubed close vertical bar space dx space equals space integral subscript negative 2 end subscript superscript 0 open vertical bar straight x cubed close vertical bar space dx space plus space integral subscript 0 superscript 1 open vertical bar straight x close vertical bar cubed space dx space equals space integral subscript negative 2 end subscript superscript 0 left parenthesis negative straight x cubed right parenthesis space dx plus integral subscript 0 superscript 1 straight x cubed space dx$
$equals negative open square brackets straight x to the power of 4 over 4 close square brackets subscript 2 superscript 0 plus open square brackets straight x to the power of 4 over 4 close square brackets subscript 0 space equals space minus open square brackets 0 minus fraction numerator left parenthesis negative 2 right parenthesis to the power of 4 over denominator 4 end fraction close square brackets plus open square brackets 1 fourth minus 0 close square brackets space equals space 4 plus 1 fourth space equals space 17 over 4$
$therefore space space space left parenthesis straight D right parenthesis space is space correct space answer.$

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).

1133 Views

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