Vector Algebra

Sponsor Area

Question

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.

Solution

The equation of curve is y² = x
Required area = $integral subscript 1 superscript 4 space straight y space dx space equals space integral subscript 1 superscript 4 square root of straight x space end root dx space equals integral subscript 1 superscript 4 straight x to the power of 1 half end exponent dx$
$equals space open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 1 superscript 4 space equals 2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 1 superscript 4 space equals space 2 over 3 open square brackets open parentheses 4 close parentheses to the power of 3 over 2 end exponent minus 1 close square brackets equals space 2 over 3 left parenthesis 8 minus 1 right parenthesis space equals space 2 over 3 cross times 7 space equals space 14 over 3 sq. space units.$

Sponsor Area

Question

Wired Faculty App

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

Solution

The equation of curve is y² = 4 x
Required area = $integral subscript 1 superscript 4 straight y space dx$
$equals space integral subscript 1 superscript 4 2 square root of straight x space dx space equals space 2 integral subscript 1 superscript 4 straight x to the power of 1 half end exponent dx equals space 2 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 1 superscript 4 space equals space 4 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 1 superscript 4 space equals space 4 over 3 open square brackets open parentheses 4 close parentheses to the power of 3 over 2 end exponent minus 1 close square brackets equals space 4 over 3 left parenthesis 8 minus 1 right parenthesis space equals space 4 over 3 cross times 7 space equals space 28 over 3 space sq. space units$

Question

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

Solution

The equation of curve is y² = 9x, which is right handed parabola.
Two lines are x = 2, x = 4.
Required area = Area ABCD
$equals space integral subscript 2 superscript 4 straight y space dx space equals space integral subscript 2 superscript 4 3 square root of straight x space dx space equals space 3 space integral subscript 2 superscript 4 straight x to the power of 1 half end exponent dx space equals space 3 open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 2 superscript 4 space equals space 3 cross times 2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 2 superscript 4 equals space 2. space open square brackets left parenthesis 4 right parenthesis to the power of 3 over 2 end exponent minus left parenthesis 2 right parenthesis to the power of 3 over 2 end exponent close square brackets space equals space 2 space left square bracket space 8 space minus square root of 8 right square bracket space equals space 2 left parenthesis 8 minus 2 square root of 2 right parenthesis equals space left parenthesis 16 minus 4 square root of 2 right parenthesis space sq. space units.$

Question

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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.

Solution

The equation of curve is y² = x - 2. which is right handed parabola with vertex at (2, 0).
Two lines are x = 4 and x = 6
Required area = Area ABCD
$equals space integral subscript 4 superscript 6 straight y space dx space equals space integral subscript 4 superscript 6 square root of straight x minus 2 end root dx equals space integral subscript 4 superscript 6 left parenthesis straight x minus 2 right parenthesis to the power of 1 half end exponent dx equals space open square brackets fraction numerator left parenthesis straight x minus 2 right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 4 superscript 6 space equals space 2 over 3 open square brackets left parenthesis straight x minus 2 right parenthesis to the power of 3 over 2 end exponent close square brackets subscript 4 superscript 6 equals space 2 over 3 open square brackets 4 to the power of 3 over 2 end exponent minus 2 to the power of 3 over 2 end exponent close square brackets space equals space space 2 over 3 left parenthesis 8 minus 2 square root of 2 right parenthesis space sq. space units$

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Vector Algebra

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.

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

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

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

WhatsApp Group

Download Android App

Ncert Book Download

Vector Algebra

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.

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

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

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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.

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

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

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