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Application of Integrals

Short Answer Type

1. Find the area of the region bounded by the curve y²= x and the lines x = 1, x = 4 and the x-axis in the first quadrant.

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Find the area of the region bounded by y² = 4 x, x = 1, x = 4 and the x-axis in the first quadrant.

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3. Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

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Find the area of the region bounded by y² = x - 2, x = 4, x = 6 and the x-axis in the first quadrant.

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5. Find the area under the given curves and given lines:
(i) y = x², x = 1, x = 2 and x-axis
(ii) y - x^4, x = 1, x = 5 and x -axis

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Find the area of the region bounded by x² = 4 y, y = 2, y = 4 and the y-axis in the first quadrant.

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7. Find the area of the region lying in the first quadrant and bounded by y = 4 x², x = 0, y = 1 and y = 4.

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8. Find the area of the region lying in the first quadrant and bounded by x² = y - 3, y = 4, y = 6 and the y-axis in the first quadrant.

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9.
Find the area of the region bounded by x² = 16 y, y = 1, y = 4 and the y-axis in the first quadrant.

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The equation of curve is
$straight x squared space equals space 16 space straight y$
Required area = $integral subscript 1 superscript 4 straight x space dy$
$equals space integral subscript 1 superscript 4 4 square root of straight y dy space equals space 4 integral subscript 1 superscript 4 straight y to the power of 1 half end exponent dy space equals space 4 open square brackets fraction numerator straight y to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 1 superscript 4 equals space 8 over 3 open square brackets straight y to the power of 3 over 2 end exponent close square brackets subscript 1 superscript 4 space equals space 8 over 3 open square brackets left parenthesis 4 right parenthesis to the power of 3 over 2 end exponent minus 1 close square brackets space equals space 8 over 3 left square bracket 8 minus 1 right square bracket space equals 8 over 3 cross times 7 space equals space 56 over 3 space sq. space units.$