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11. Find the area of the region bounded by the curve

straight y space equals space 2 square root of 1 minus straight x squared end root

and x-axis. Draw a rough sketch..

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12. Calculate the area bounded by the parabola y² = 4 a x and its latus rectum.

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13.

Give the rough sketch of the curve y² = x and the line x = 4 and find the area between the curve and the line.

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14.

Find the area bounded between the curve y² = x and the line x = 3. Draw the rough sketch also.

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15. The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

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16.

Find the area of the region bounded by the curve y = x² and the line y = 4.

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17.

Find the area of the region bounded by the curve y² = 4x and the line x = 3.

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18.

Using definite integrals, find the area of the circle x² + y² = a².

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19. Find the area of the region bounded by the ellipse

straight x squared over 9 plus straight y squared over 4 space equals space 1.

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The equation of ellipse is

straight x squared over 16 plus straight y squared over 9 space equals 1

straight y squared over 9 space equals space 1 minus straight x squared over 16

straight y squared space equals 9 over 16 left parenthesis 16 minus straight x squared right parenthesis

therefore space space space space space space straight y space equals space 3 over 4 square root of 16 minus straight x squared end root

...(1)

(in the first quadrant)
The ellipse is symmetrical about both the axes.
∴ required area = 4 (area AOB)

equals space 4 integral subscript 0 superscript 4 straight y space dx space equals space 4 integral subscript 0 superscript 4 3 over 4 square root of 16 minus straight x squared end root dx

[

because space of space left parenthesis 1 right parenthesis

]

equals 3 integral subscript 0 superscript 4 square root of left parenthesis 4 right parenthesis squared minus straight x squared end root dx space equals space 3 open square brackets fraction numerator straight x square root of 16 minus straight x squared end root over denominator 2 end fraction plus fraction numerator left parenthesis 4 right parenthesis squared over denominator 2 end fraction sin to the power of negative 1 end exponent open parentheses straight x over 4 close parentheses close square brackets subscript 0 superscript 4 equals space 3 open square brackets open curly brackets fraction numerator 4 square root of 16 minus 16 end root over denominator 2 end fraction plus 8 space sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis close curly brackets minus open curly brackets 0 plus 8 space sin to the power of negative 1 end exponent 0 close curly brackets close square brackets equals space 3 open square brackets open parentheses 0 plus 8 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 8 cross times 0 right parenthesis close square brackets space equals space 3 cross times 8 cross times straight pi over 2 space equals space 12 space straight pi space sq. space units. space

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