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.
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.
Find the area of the region bounded by the ellipse `x^2/16 + y^2/9 = 1.`
Find the area enclosed between the parabola y2 = 4ax and the line y = mx
Find the area of the smaller region bounded by the ellipse `x^2/9 + y^2/4` and the line `x/3 + y/2 = 1`
Find the area of the smaller region bounded by the ellipse `x^2/a^2 + y^2/b^2 = 1` and the line `x/a + y/b = 1`
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 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: 2y = 5x + 7, x = 2, x = 8
Choose the correct alternative :
Area of the region bounded by the curve x2 = y, the X-axis and the lines x = 1 and x = 3 is _______.
Choose the correct alternative :
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _____.
Using definite integration, area of the circle x2 + y2 = 49 is _______.
If the curve, under consideration, is below the X-axis, then the area bounded by curve, X-axis and lines x = a, x = b is positive.
Solve the following :
Find the area of the region bounded by the curve y = x2 and the line y = 10.
Area of the region bounded by the curve x = y2, the positive Y axis and the lines y = 1 and y = 3 is ______
Choose the correct alternative:
Area of the region bounded by x = y4, y = 1 and y = 5 and the Y-axis lying in the first quadrant is ______
State whether the following statement is True or False:
The area bounded by the curve y = f(x) lies on the both sides of the X-axis is `|int_"a"^"b" "f"(x) "d"x| + |int_"b"^"c" "f"(x) "d"x|`
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 area of the region bounded by the curve y = – 4x, the X-axis and the lines x = – 1 and x = 2
Area enclosed between the curve y2(4 - x) = x3 and line x = 4 above X-axis is ______.
The equation of curve through the point (1, 0), if the slope of the tangent to t e curve at any point (x, y) is `(y - 1)/(x^2 + x)`, is
The area included between the parabolas y2 = 4a(x +a) and y2 = 4b(x – a), b > a > 0, is
The area of the region bounded by the curve y = sin x and the x-axis in [–π, π] is ______.
Area bounded by y = sec2x, x = `π/6`, x = `π/3` and x-axis is ______.
The area (in sq. units) of the region {(x, y) : y2 ≥ 2x and x2 + y2 ≤ 4x, x ≥ 0, y ≥ 0} is ______.
The area bounded by the curve, y = –x, X-axis, x = 1 and x = 4 is ______.
