Advertisements
Advertisements
प्रश्न
Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the Y-axis. Hence, obtain its area using integration.
Advertisements
उत्तर
2x + y = 8, y = 2, y = 4
Required Area = Area of ABDE + Area of BCD
= `int_0^2 (4 - 2)dx + int_2^3 {(8 - 2x) - 2}dx`
= `[2x]_0^2 + [6x - x^2]_2^3`
= 4 + [18 – 9 – 12 + 4]
= 4 + 1
= 5 sq. units.
APPEARS IN
संबंधित प्रश्न
Draw a rough sketch of the curve and find the area of the region bounded by curve y2 = 8x and the line x =2.
Find the area lying above the x-axis and under the parabola y = 4x − x2.
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.
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 x2 = 4ay and its latusrectum.
Find the area of the region bounded by the curve \[a y^2 = x^3\], the y-axis and the lines y = a and y = 2a.
Find the area of the region \[\left\{ \left( x, y \right): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \leq \frac{x}{a} + \frac{y}{b} \right\}\]
Find the area common to the circle x2 + y2 = 16 a2 and the parabola y2 = 6 ax.
OR
Find the area of the region {(x, y) : y2 ≤ 6ax} and {(x, y) : x2 + y2 ≤ 16a2}.
Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).
Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.
Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).
Find the area enclosed by the curve \[y = - x^2\] and the straight line x + y + 2 = 0.
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 in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2= 32.
Find the area of the region enclosed between the two curves x2 + y2 = 9 and (x − 3)2 + y2 = 9.
The area bounded by the curve y = loge x and x-axis and the straight line x = e is ___________ .
The area bounded by the parabola x = 4 − y2 and y-axis, in square units, is ____________ .
The area bounded by the curve y = x4 − 2x3 + x2 + 3 with x-axis and ordinates corresponding to the minima of y 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 ________ .
Find the area of the region above the x-axis, included between the parabola y2 = ax and the circle x2 + y2 = 2ax.
The area of the region bounded by the y-axis, y = cosx and y = sinx, 0 ≤ x ≤ `pi/2` is ______.
The area of the region bounded by the curve y = `sqrt(16 - x^2)` and x-axis is ______.
Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is ______.
What is the area of the region bounded by the curve `y^2 = 4x` and the line `x` = 3.
Smaller area bounded by the circle `x^2 + y^2 = 4` and the line `x + y = 2` is.
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 ______.
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 ______.
Find the area of the following region using integration ((x, y) : y2 ≤ 2x and y ≥ x – 4).
