Advertisements
Advertisements
प्रश्न
Find the area bounded by the curve y = 2cosx and the x-axis from x = 0 to x = 2π
उत्तर
Given equation of the curve is y = 2 cos x
∴ Area of the shaded region = `int_0^(2pi) 2 cos x "d"x`
= `int_0^(pi/2) 2 cos x "d"x + int_(pi/2)^((3pi)/2) |2 cos x|"d"x + int_((3pi)/2)^(2pi) 2 cos x "d"x`
= `2[sin x]_0^(pi/2) + |[2 sin x]_(pi/2)^((3pi)/2)| + 2[sin x]_((3pi)/2)^(2pi)`
= `2[sin pi/2 - sin 0] + |2(sin (3pi)/2 - sin pi/2)| + 2[sin 2pi - sin (3pi)/2]`
= `2(1) + |2(-1 - 1)| + 2(0 + 1)`
= 2 + 4 + 2
= 8 sq.units
APPEARS IN
संबंधित प्रश्न
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum.
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.
Find the area bounded by the curve y = sin x between x = 0 and x = 2π.
Find the area of the region bounded by the parabola y2 = 4ax and the line x = a.
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.
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.
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.
Using integration, find the area of the triangular region, the equations of whose sides are y = 2x + 1, y = 3x+ 1 and x = 4.
Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.
Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.
Find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.
Using integration, find the area of the following region: \[\left\{ \left( x, y \right) : \frac{x^2}{9} + \frac{y^2}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2} \right\}\]
The closed area made by the parabola y = 2x2 and y = x2 + 4 is __________ .
The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is given by
The area bounded by the curve y = 4x − x2 and the x-axis is __________ .
Area bounded by the curve y = x3, the x-axis and the ordinates x = −2 and x = 1 is ______.
The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \[\frac{\pi}{2}\] is _________ .
Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3, is
Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3.
Find the area of the curve y = sin x between 0 and π.
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.
Find the area of region bounded by the triangle whose vertices are (–1, 1), (0, 5) and (3, 2), using integration.
Find the area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5.
The area of the region bounded by the y-axis, y = cosx and y = sinx, 0 ≤ x ≤ `pi/2` is ______.
Using integration, find the area of the region bounded between the line x = 4 and the parabola y2 = 16x.
Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x2, 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.
Find the area bounded by the curve y = |x – 1| and y = 1, using integration.
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 ______.
