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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

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

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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.

Consider the given equation.
$straight y equals square root of 5 minus straight x squared end root$
This equation represents a semicircle with centre at the origin and radius = $square root of 5 space units$
Given that the region is bounded by the above semicircle and the line $straight y equals open vertical bar straight x minus 1 close vertical bar$
Let us find the point of intersection of the given curve meets the line $straight y equals open vertical bar straight x minus 1 close vertical bar$
$rightwards double arrow square root of 5 minus straight x squared end root space equals space open vertical bar straight x minus 1 close vertical bar$
Squaring both the sides, we have,

$5 minus straight x squared space equals open vertical bar straight x minus 1 close vertical bar squared rightwards double arrow 5 minus straight x squared space equals space straight x squared plus 1 minus 2 straight x rightwards double arrow 2 straight x squared minus 2 straight x minus 5 plus 1 space equals space 0 rightwards double arrow 2 straight x squared minus 2 straight x minus 4 space equals space 0 rightwards double arrow straight x squared minus straight x minus 2 space equals 0 rightwards double arrow space straight x squared minus 2 straight x plus straight x minus 2 space equals 0 rightwards double arrow straight x left parenthesis straight x minus 2 right parenthesis plus 1 left parenthesis straight x minus 2 right parenthesis equals 0 rightwards double arrow left parenthesis straight x plus 1 right parenthesis thin space left parenthesis straight x minus 2 right parenthesis space equals space 0 rightwards double arrow straight x space equals negative 1 comma space straight x space equals space 2 When space straight x space equals space minus 1 comma space space straight y space equals space 2 When space straight x space equals space 2 comma space space straight y space equals space 1$
Consider the following figure
Thus the intersection points are 1,2 and 2,1 ( ) ( ) Consider the following sketch of the bounded region.

Required Area, $straight A space equals space integral subscript negative 1 end subscript superscript 2 left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis dx space$
$equals integral subscript negative 1 end subscript superscript 1 space open square brackets square root of 5 minus straight x squared end root plus left parenthesis straight x minus 1 right parenthesis close square brackets dx space plus space integral subscript 1 superscript 2 space open square brackets square root of 5 minus straight x squared end root minus left parenthesis straight x minus 1 right parenthesis close square brackets dx equals integral subscript negative 1 end subscript superscript 1 space square root of 5 minus straight x squared end root dx plus integral subscript negative 1 end subscript superscript 1 xdx space minus integral subscript negative 1 end subscript superscript 1 dx space plus space integral subscript 1 superscript 2 square root of 5 minus straight x squared end root dx minus integral subscript 1 superscript 2 xdx plus integral subscript 1 superscript 2 dx equals open square brackets straight x over 2 square root of 5 minus straight x squared end root plus 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 5 end fraction close parentheses close square brackets subscript negative 1 end subscript superscript 1 space plus space open parentheses straight x squared over 2 close parentheses subscript negative 1 end subscript superscript 1 space minus left parenthesis straight x right parenthesis subscript negative 1 end subscript superscript 1 plus open square brackets straight x over 2 square root of 5 minus straight x squared end root plus 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator square root of 5 end fraction close parentheses close square brackets subscript 1 superscript 2 minus open parentheses straight x squared over 2 close parentheses subscript 1 superscript 2 space plus left parenthesis straight x right parenthesis subscript 1 superscript 2 equals 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 5 end fraction close parentheses plus 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator 2 over denominator square root of 5 end fraction close parentheses minus 1 half$

$Required space Area space equals open square brackets 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 5 end fraction close parentheses plus 5 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator 2 over denominator square root of 5 end fraction close parentheses minus 1 half close square brackets sq. space units$