Advertisements
Advertisements
प्रश्न
Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.
उत्तर
Here, we are given y = `sqrt("a"^2 - x^2)`
⇒ y2 = a2 – x2
⇒ x2 + y2 = a2
Area of the shaded region
= `2[(1)^(3/2) - 0] - 3/2[(1)^2 - 0]`
= `[x/2 sqrt("a"^2 - x^2) + "a"^2/2 sin^-1 x/"a"]_0^"a"`
= `["a"/2 sqrt("a"^2 - "a"^2) + "a"^2/2 sin^-1 "a"/"a" - 0 - 0]`
= `"a"^2/2 sin^-1 (1)`
= `"a"^2/2 * pi/2`
= `(pi"a"^2)/4`
Hence, the required area = `(pi"a"^2)/4` sq.units.
APPEARS IN
संबंधित प्रश्न
Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y
Using integration find the area of the triangular region whose sides have the equations y = 2x +1, y = 3x + 1 and x = 4.
Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is
A. 2 (π – 2)
B. π – 2
C. 2π – 1
D. 2 (π + 2)
Find the area bounded by curves {(x, y) : y ≥ x2 and y = |x|}.
Choose the correct answer The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is
A. `4/3 (4pi - sqrt3)`
B. `4/3 (4pi + sqrt3)`
C. `4/3 (8pi - sqrt3)`
D.`4/3 (4pi + sqrt3)`
The area bounded by the y-axis, y = cos x and y = sin x when 0 <= x <= `pi/2`
(A) 2 ( 2 −1)
(B) `sqrt2 -1`
(C) `sqrt2 + 1`
D. `sqrt2`
Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.
The area enclosed between the curves y = loge (x + e), x = loge \[\left( \frac{1}{y} \right)\] and the x-axis is _______ .
The area between x-axis and curve y = cos x when 0 ≤ x ≤ 2 π is ___________ .
Area enclosed between the curve y2 (2a − x) = x3 and the line x = 2a above x-axis is ___________ .
Area lying between the curves y2 = 4x and y = 2x is
The area of the region included between the parabolas y2 = 16x and x2 = 16y, is given by ______ sq.units
Find the area enclosed between y = cos x and X-axis between the lines x = `pi/2` and x ≤ `(3pi)/2`
Find the area of the ellipse `x^2/1 + y^2/4` = 1, in first quadrant
Find the area of sector bounded by the circle x2 + y2 = 25, in the first quadrant−
Find the area enclosed between the X-axis and the curve y = sin x for values of x between 0 to 2π
Find the area of the region lying between the parabolas 4y2 = 9x and 3x2 = 16y
Find the area enclosed between the circle x2 + y2 = 9, along X-axis and the line x = y, lying in the first quadrant
Find the area enclosed by the curve x = 3 cost, y = 2 sint.
Find the area of the region included between the parabola y = `(3x^2)/4` and the line 3x – 2y + 12 = 0.
The value of a for which the area between the curves y2 = 4ax and x2 = 4ay is 1 sq.unit, is ______.
Using Integration, find the area of triangle whose vertices are (– 1, 1), (0, 5) and (3, 2).
Find the area cut off from the parabola 4y = 3x2 by the line 2y = 3x + 12.
