Advertisements
Advertisements
प्रश्न
Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2.
Advertisements
उत्तर
The equations of the given lines are
y = 2 + x .....(1)
y = 2 – x .....(2)
x = 2 .....(3)
Solving (1), (2) and (3) in pairs, we obtain the coordinates of the point of intersection as A(0, 2), B(2, 4) and C(2, 0).
The graphs of these lines are drawn as shown in the figure below.
Here, the shaded region represents the area bounded by the given lines.
∴ Required area = Area of the region ABCA
= Area of region ACDA + Area of region ABDA
`=∫_0^2xAC dy+∫_2^4xAB dy`
`=∫_0^2(2−y)dy+∫_2^4(y−2)dy`
`=(2y-y^2/2)_0^2+(Y^2/2-2y)_2^4`
=[(4−2)−(0−0)]+[(8−8)−(2−4)]
=2+2
=4 square units
Thus, the area of the region bounded by the given lines is 4 square 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 parabola y2 = 16x and the line x = 3.
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.
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
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.
Determine the area under the curve y = `sqrt(a^2-x^2)` included between the lines x = 0 and x = a.
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.
Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =\[\frac{\pi}{3}\] are in the ratio 2 : 3.
Compare the areas under the curves y = cos2 x and y = sin2 x between x = 0 and x = π.
If the area enclosed by the parabolas y2 = 16ax and x2 = 16ay, a > 0 is \[\frac{1024}{3}\] square units, find the value of a.
The area included between the parabolas y2 = 4x and x2 = 4y is (in square units)
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 ________ .
Area bounded by parabola y2 = x and straight line 2y = x is _________ .
The area bounded by the curve y = 4x − x2 and the x-axis is __________ .
Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3, is
The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is ______.
Draw a rough sketch of the given curve y = 1 + |x +1|, x = –3, x = 3, y = 0 and find the area of the region bounded by them, using integration.
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 = x + 1 and the lines x = 2 and x = 3 is ______.
The curve x = t2 + t + 1,y = t2 – t + 1 represents
Area of the region bounded by the curve y = |x + 1| + 1, x = –3, x = 3 and y = 0 is
Find the area of the region bounded by `x^2 = 4y, y = 2, y = 4`, and the `y`-axis in the first quadrant.
What is the area of the region bounded by the curve `y^2 = 4x` and the line `x` = 3.
Let the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) 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 ______.
Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the Y-axis. Hence, obtain its area using integration.
Using integration, find the area of the region bounded by y = mx (m > 0), x = 1, x = 2 and the X-axis.
