Advertisements
Advertisements
प्रश्न
Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py
Advertisements
उत्तर
We are given that: x2 = 2py ......(i)
And y2 = 2px ......(ii)
From equation (i)
We get y = `x^2/(2"p")`
Putting the value of y in equation (ii)
We have `(x^2/(2"p"))` = 2px
⇒ `x^4/(4"p"^2)` = 2px
⇒ x4 = 8p3x
⇒ x4 – 8p3x = 0
⇒ x(x3 – 8p3) = 0
∴ x = 0, 2p
Required area = Area of the region (OCBA – ODBA)
= `int_0^(2"p") sqrt(2"p"x) "d"x - int_0^(2"p") x^2/(2"p") "d"x`
= `sqrt(2"p") * 2/3 [x^(3/2)]_0^(2"p") - 1/(2"p") * 1/3 [x^3]_0^(2"p")`
= `(2sqrt(2))/3 sqrt("p") [(2"p")^(3/2) - 0] - 1/(6"p") [(2"p")^3 - 0]`
= `(2sqrt(2))/3 sqrt("p") * 2sqrt(2) "p"^(3/2) - 1/(6"p") * 8"p"^3`
= `8/3 * "p"^2 - 8/6 "p"^2`
= `8/6 "p"^2`
= `4/3 "p"^2` sq.units
Hence, the required area = `4/3 "p"^2` sq.units
APPEARS IN
संबंधित प्रश्न
Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis
triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.
Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.
Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2.
Using integration, find the area of the region bounded by the line 2y = 5x + 7, x-axis and the lines x = 2 and x = 8.
Using definite integrals, find the area of the circle x2 + y2 = a2.
Using integration, find the area of the region bounded by the following curves, after making a rough sketch: y = 1 + | x + 1 |, x = −2, x = 3, y = 0.
Find the area of the region bounded by x2 + 16y = 0 and its latusrectum.
Find the area of the region common to the parabolas 4y2 = 9x and 3x2 = 16y.
Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.
Find the area of the region in the first quadrant enclosed by x-axis, the line y = \[\sqrt{3}x\] and the circle x2 + y2 = 16.
Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).
Sketch the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 1. Also, find the area of this region.
Using integration, find the area of the triangle ABC coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).
Find the area of the region bounded by y = | x − 1 | and y = 1.
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2.
Find the area enclosed by the curves y = | x − 1 | and y = −| x − 1 | + 1.
Find the area enclosed by the curves 3x2 + 5y = 32 and y = | x − 2 |.
The area bounded by y = 2 − x2 and x + y = 0 is _________ .
The area of the region \[\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}\] is __________ .
The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is (b −1) sin (3b + 4). Then, f (x) is __________ .
Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.
Using integration, find the area of the smaller region bounded by the ellipse `"x"^2/9+"y"^2/4=1`and the line `"x"/3+"y"/2=1.`
The area enclosed by the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 is equal to ______.
Find the area of the region included between y2 = 9x and y = x
Find the area of region bounded by the line x = 2 and the parabola y2 = 8x
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Find the area bounded by the curve y = sinx between x = 0 and x = 2π.
The area of the region bounded by the curve y = sinx between the ordinates x = 0, x = `pi/2` and the x-axis is ______.
The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is ______.
The area of the region bounded by the line y = 4 and the curve y = x2 is ______.
Find the area bounded by the curve y = |x – 1| and y = 1, using integration.
For real number a, b (a > b > 0),
let Area `{(x, y): x^2 + y^2 ≤ a^2 and x^2/a^2 + y^2/b^2 ≥ 1}` = 30π
Area `{(x, y): x^2 + y^2 ≥ b^2 and x^2/a^2 + y^2/b^2 ≤ 1}` = 18π.
Then the value of (a – b)2 is equal to ______.
Area (in sq.units) of the region outside `|x|/2 + |y|/3` = 1 and inside the ellipse `x^2/4 + y^2/9` = 1 is ______.
Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = `π/4` and x = `β > π/4` is `(βsinβ + π/4 cos β + sqrt(2)β)`. Then `f(π/2)` is ______.
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 – 3x2 – 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______.
Find the area of the smaller region bounded by the curves `x^2/25 + y^2/16` = 1 and `x/5 + y/4` = 1, using integration.
