Advertisements
Advertisements
Question
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9 = 1.`
Advertisements
Solution
The given ellipse is `x^2/4 + y^2/9 = 1`
Since the given curve is symmetrical about both axes.
∴ Area of ellipse = 4 areas (OABO)
∴ Requied area = 4 Area (OABO) = `4 int_0^3 x dy` ...[by taking horizontal strips]
`4 int_0^3 2/3 sqrt (9 - y^2) dx` `...[x^2/4 + y^2/9 = 1 ⇒ x^2/4 = 1 - y^2/9 ⇒ x = 2/3 sqrt (9 - y^2) (∵ x > 0)]`
`= 4 xx 2/3 [y/2 sqrt (9 - y^2) + 9/2 sin^-1 y/3]_0^3`
`= 4 xx 2/3 [(3/2 (0) + 9/2 sin^-1 (1)) - (0 - 0)]`
`= 4 xx 2/3 [9/2 (pi/2)]`
`= 4 xx (3pi)/2`
= 6π square 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}.
Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is ______.
Find the area under the given curve and given line:
y = x2, x = 1, x = 2 and x-axis
Find the area under the given curve and given line:
y = x4, x = 1, x = 5 and x-axis
Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 and x-axis
Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`
Draw a rough sketch and find the area bounded by the curve x2 = y and x + y = 2.
Find the area of the region.
{(x,y) : 0 ≤ y ≤ x2 , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .
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
Find the area of the region bounded by the following curves, the X-axis and the given lines:
y = x2 + 1, x = 0, x = 3
Find the area of the region bounded by the following curve, the X-axis and the given line:
y = 2 – x2, x = –1, x = 1
Choose the correct alternative :
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _____.
State whether the following is True or False :
The area bounded by the curve y = f(x), X-axis and lines x = a and x = b is `|int_"a"^"b" f(x)*dx|`.
Solve the following :
Find the area of the region bounded by the curve y = x2 and the line y = 10.
Find the area of the region bounded by y = x2, the X-axis and x = 1, x = 4.
Choose the correct alternative:
Area of the region bounded by the curve y = x3, x = 1, x = 4 and the X-axis is ______
Choose the correct alternative:
Area of the region bounded by the parabola y2 = 25x and the lines x = 5 is ______
The area of the shaded region bounded by two curves y = f(x), and y = g(x) and X-axis is `int_"a"^"b" "f"(x) "d"x + int_"a"^"b" "g"(x) "d"x`
The area of the region lying in the first quadrant and bounded by the curve y = 4x2, and the lines y = 2 and y = 4 is ______
Find the area of the region bounded by the curve 4y = 7x + 9, the X-axis and the lines x = 2 and x = 8
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 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 bounded by the curve xy = 4, X-axis between x = 1, x = 5 is ______.
Area enclosed between the curve y2(4 - x) = x3 and line x = 4 above X-axis is ______.
Area under the curve `y=sqrt(4x+1)` between x = 0 and x = 2 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 ______.
Which equation below represents a parabola that opens upward with a vertex at (0, – 5)?
If a2 + b2 + c2 = – 2 and f(x) = `|(1 + a^2x, (1 + b^2)x, (1 + c^2)x),((1 + a^2)x, 1 + b^2x, (1 + c^2)x),((1 + a^2)x, (1 + b^2)x, 1 + c^2x)|` then f(x) is a polynomial of degree
The area of the region bounded by the curve y = x2, x = 0, x = 3, and the X-axis is ______.
Area bounded by the curves y = `"e"^(x^2)`, the x-axis and the lines x = 1, x = 2 is given to be α square units. If the area bounded by the curve y = `sqrt(ℓ "n"x)`, the x-axis and the lines x = e and x = e4 is expressed as (pe4 – qe – α), (where p and q are positive integers), then (p + q) is ______.
The area bounded by the x-axis and the curve y = 4x – x2 – 3 is ______.
The area bounded by the curve, y = –x, X-axis, x = 1 and x = 4 is ______.
