Advertisements
Advertisements
Question
Find the area bounded by the circle x2 + y2 = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.
Advertisements
Solution
The area bounded by the circle x2 + y2 = 16 , x = `sqrt3 y = x` , and the x-axis is the area OAB.
Solving x2 + y2 = 16 , x = `sqrt3 y = x` we have
`(sqrt3y)^2 + y^2 = 16`
⇒3y2 + y2 = 16
⇒4y2 = 16
⇒y2 = 4
⇒ y = 2 (In the first quadrant, y is positive)
When y = 2, x = `2sqrt3`
So, the point of intersection of the given line and circle in the first quadrant is `(2sqrt3, 2)`
The graph of the given line and cirlce is shown below:
Required area = Area of the shaded region = Area OABO = Area OCAO + Area ACB
Area OCAO = `1/2 xx 2sqrt3 xx 2 = 2sqrt3` sq units
Area ABC = `int_(2sqrt3)^4 ydx`
= `int_(2sqrt3)^4 sqrt(16 - x^2) dx`
`= [x/2 sqrt(16 - x^2) + 16/2 sin^(-1) x/4]_(2sqrt3)^4`
`=[(0 + 8sin^(-1) 1) - ((2sqrt3)/3 xx 2 + 8 xx sin^(-1) sqrt3/2)]`
`= 8 xx pi/2 - 2sqrt3 - 8 xx pi/3`
= `((4pi)/3 - 2sqrt3)` sq unit
∴ Required area = `((4pi)/3 - 2sqrt3) + 2sqrt3 = (4pi)/3` sq units
APPEARS IN
RELATED QUESTIONS
Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9 = 1.`
Find the area of the region in the first quadrant enclosed by x-axis, line x = `sqrt3` y and the circle x2 + y2 = 4.
Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is ______.
Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4
Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B (4, 5) and C (6, 3).
Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x - 2y + 12 = 0.
Find the area of the region bounded by the parabola y2 = 16x and the line x = 4.
Find the area of the region bounded by the following curves, the X-axis, and the given lines:
y = `sqrt(6x + 4), x = 0, x = 2`
Find the area of the region bounded by the following curves, the X-axis and the given lines: 2y + x = 8, x = 2, x = 4
Choose the correct alternative:
Using the definite integration area of the circle x2 + y2 = 16 is ______
The area of the region bounded by y2 = 25x, x = 1 and x = 2 the X axis is ______
Find the area of the region bounded by the curve y = `sqrt(9 - x^2)`, X-axis and lines x = 0 and x = 3
Find the area of the region bounded by the curve y = `sqrt(2x + 3)`, the X axis and the lines x = 0 and x = 2
Find the area of the region bounded by the curve y = (x2 + 2)2, the X-axis and the lines x = 1 and x = 3
Find area of the region bounded by 2x + 4y = 10, y = 2 and y = 4 and the Y-axis lying in the first quadrant
Find area of the region bounded by the parabola x2 = 4y, the Y-axis lying in the first quadrant and the lines y = 3
Find the area of the region bounded by the curve x = `sqrt(25 - y^2)`, the Y-axis lying in the first quadrant and the lines y = 0 and y = 5
Find the area of the circle x2 + y2 = 62
The area enclosed between the curve y = loge(x + e) and the coordinate axes is ______.
The ratio in which the area bounded by the curves y2 = 8x and x2 = 8y is divided by the line x = 2 is ______
Area under the curve `y=sqrt(4x+1)` between x = 0 and x = 2 is ______.
The area enclosed by the parabolas x = y2 - 1 and x = 1 - y2 is ______.
The area bounded by the X-axis, the curve y = f(x) and the lines x = 1, x = b is equal to `sqrt("b"^2 + 1) - sqrt(2)` for all b > 1, then f(x) is ______.
The area enclosed by the parabola x2 = 4y and its latus rectum is `8/(6m)` sq units. Then the value of m is ______.
