Advertisements
Advertisements
Question
Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.
Solution
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
RELATED QUESTIONS
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 lying between the parabolas y2 = 4ax and x2 = 4ay.
Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y
Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y 2 = 1
Using integration finds the area of the region bounded by the triangle whose vertices are (–1, 0), (1, 3) and (3, 2).
Using integration find the area of the triangular region whose sides have the equations y = 2x +1, y = 3x + 1 and x = 4.
Area lying between the curve y2 = 4x and y = 2x is
A. 2/3
B. 1/3
C. 1/4
D. 3/4
Using the method of integration find the area bounded by the curve |x| + |y| = 1 .
[Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1 and
– x – y = 1].
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 _______ .
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
Solve the following :
Find the area of the region lying between the parabolas :
y2 = 4x and x2 = 4y
The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ sq.units
Find the area of the ellipse `x^2/1 + y^2/4` = 1, in first quadrant
Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration
Find the area of the region included between y = x2 + 5 and the line y = x + 7
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 enclosed between 3y = x2, X-axis and x = 2 to x = 3.
Find the area cut off from the parabola 4y = 3x2 by the line 2y = 3x + 12.
