Advertisements
Advertisements
प्रश्न
Find the area of the region bounded by the curves y2 = 9x, y = 3x
Advertisements
उत्तर
We have, y2 = 9x, y = 3x
Solving the two equations,
We have (3x)2 = 9x
⇒ 9x2 – 9x = 0
⇒ 9x(x – 1) = 0
∴ x = 0, 1
Area of the shaded region
= ar (region OAB) – ar (ΔOAB)
= `- int_0^1 y_1 * "d"x`
= `int_0^1 sqrt(9x) "d"x - int_0^1 3x "d"x`
= `3 int_0^1 sqrt(x) "d"x - 3 int_0^1 x "d"x`
= `3 xx 2/3 [x^(3/2)]_0^1 - 3[x^2/2]_0^1`
= `2[(1)^(3/2) - 0] - 3/2 [(1)^2 - 0]`
= `2(1) - 3/2 (1)`
= `2 - 3/2`
= `1/2` sq.units
Hence, the required area = `1/2` sq.units
APPEARS IN
संबंधित प्रश्न
Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y − 2.
Find the area of the region bounded by the curve x2 = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.
Area bounded by the curve y = x3, the x-axis and the ordinates x = –2 and x = 1 is ______.
Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.
Draw a rough sketch of the graph of the curve \[\frac{x^2}{4} + \frac{y^2}{9} = 1\] and evaluate the area of the region under the curve and above the x-axis.
Draw a rough sketch of the graph of the function y = 2 \[\sqrt{1 - x^2}\] , x ∈ [0, 1] and evaluate the area enclosed between the curve and the x-axis.
Draw a rough sketch of the curve y = \[\frac{\pi}{2} + 2 \sin^2 x\] and find the area between x-axis, the curve and the ordinates x = 0, x = π.
Draw a rough sketch of the curve \[y = \frac{x}{\pi} + 2 \sin^2 x\] and find the area between the x-axis, the curve and the ordinates x = 0 and x = π.
Compare the areas under the curves y = cos2 x and y = sin2 x between x = 0 and x = π.
Find the area of the region bounded by y =\[\sqrt{x}\] and y = x.
Prove that the area common to the two parabolas y = 2x2 and y = x2 + 4 is \[\frac{32}{3}\] sq. units.
Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).
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 enclosed by the parabola x2 = y and the line y = x + 2.
Find the area bounded by the lines y = 4x + 5, y = 5 − x and 4y = x + 5.
Find the area enclosed by the curves 3x2 + 5y = 32 and y = | x − 2 |.
In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x − x2 and y = x2− x?
Find the area of the figure bounded by the curves y = | x − 1 | and y = 3 −| x |.
Find the area bounded by the parabola x = 8 + 2y − y2; the y-axis and the lines y = −1 and y = 3.
If the area above the x-axis, bounded by the curves y = 2kx and x = 0, and x = 2 is \[\frac{3}{\log_e 2}\], then the value of k is __________ .
The area bounded by the parabola x = 4 − y2 and y-axis, in square units, is ____________ .
The area bounded by the curves y = sin x between the ordinates x = 0, x = π and the x-axis is _____________ .
The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .
The area of the region \[\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}\] is __________ .
The ratio of the areas between the curves y = cos x and y = cos 2x and x-axis from x = 0 to x = π/3 is ________ .
The area bounded by the curve y2 = 8x and x2 = 8y is ___________ .
The area bounded by the parabola y2 = 8x, the x-axis and the latusrectum is ___________ .
The area bounded by the curve y = x |x| and the ordinates x = −1 and x = 1 is given by
Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x
Find the area of the curve y = sin x between 0 and π.
Find the area bounded by the curve y = 2cosx and the x-axis from x = 0 to x = 2π
The area of the region bounded by the circle x2 + y2 = 1 is ______.
The area bounded by `y`-axis, `y = cosx` and `y = sinx, 0 ≤ x - (<pi)/2` is
The area enclosed by y2 = 8x and y = `sqrt(2x)` that lies outside the triangle formed by y = `sqrt(2x)`, x = 1, y = `2sqrt(2)`, is equal to ______.
The area (in sq.units) of the region A = {(x, y) ∈ R × R/0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤x2 + 3x} 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 ______.
Let g(x) = cosx2, f(x) = `sqrt(x)`, and α, β (α < β) be the roots of the quadratic equation 18x2 – 9πx + π2 = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is ______.
