Advertisements
Advertisements
Question
Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.
Advertisements
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
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
Find the area of the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 3
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.
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)
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].
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.
Find the area included between the parabolas y2 = 4ax and x2 = 4by.
The area enclosed between the curves y = loge (x + e), x = loge \[\left( \frac{1}{y} \right)\] and the x-axis is _______ .
Solve the following :
Find the area of the region lying between the parabolas y2 = 4x and x2 = 4y.
The area of the region included between the parabolas y2 = 16x and x2 = 16y, is given by ______ sq.units
The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ 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 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 of the region included between the parabola y = `(3x^2)/4` and the line 3x – 2y + 12 = 0.
Find the area of a minor segment of the circle x2 + y2 = a2 cut off by the line x = `"a"/2`
Draw a rough sketch of the curve y = `sqrt(x - 1)` in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.
Area lying between the curves `y^2 = 4x` and `y = 2x`
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.
