Advertisements
Advertisements
प्रश्न
Find the area of the region bounded by the ellipse `x^2/16 + y^2/9 = 1.`
Advertisements
उत्तर
Given equation of ellipse `x^2/16 + y^2/9 = 1`
The given ellipse is symmetric about both axes and has identical x and y axes.
`= y^2/9 = 1 - x^2/16`
`= y = pm 3/4 (sqrt(16 - x^2))`
Area enclosed by the ellipse = 4(Area of sector) = 4(Area OAC)
Ellipse in the first quadrant `= 4 int_0^4 y dx = int_0^4 3/4 sqrt(16 - x^2) dx`
Let `x = 4 sin theta ; dx = 4 cos theta d theta`
Hence, when x = 0, `theta = 0 ;` when x = 4, `theta = pi/2`
Required Area `= (4 xx 3)/4 int_0^(pi//2) sqrt(16 - 16 sin^2 theta). 4 cos theta d theta.`
`= 3 int_0^(pi/2) 4sqrt(1 - sin^2 theta). 4 cos theta d theta`
`= 48 int_0^(pi/2) cos^2 theta d theta`
`= 24 int_0^(pi/2) (1 + cos 2 theta)d theta`
`= 24 [theta + (sin 2 theta)/2]_0^(pi/2)`
`= 12π square unit
APPEARS IN
संबंधित प्रश्न
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.
Find the area of the region bounded by the curve y2 = 4x and the line x = 3
Find the area under the given curve and given line:
y = x4, x = 1, x = 5 and x-axis
Find the area between the curves y = x and y = x2
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 bounded by the circle x2 + y2 = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.
Using integration, find the area of the region {(x, y) : x2 + y2 ≤ 1 ≤ x + y}.
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:
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
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _______.
Fill in the blank :
The area of the region bounded by the curve x2 = y, the X-axis and the lines x = 3 and x = 9 is _______.
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:
Using the definite integration area of the circle x2 + y2 = 16 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|`
State whether the following statement is True or False:
The equation of the area of the circle is `x^2/"a"^2 + y^2/"b"^2` = 1
The area of the region bounded by the curve y2 = x and the Y axis in the first quadrant and lines y = 3 and y = 9 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 parabola y2 = 25x and the line x = 5
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 y = `sqrt(36 - x^2)`, the X-axis lying in the first quadrant and the lines x = 0 and x = 6
If `int_0^(pi/2) log (cos x) "dx" = - pi/2 log 2,` then `int_0^(pi/2) log (cosec x)`dx = ?
The area of the region bounded by the curve y = 4x3 − 6x2 + 4x + 1 and the lines x = 1, x = 5 and X-axis is ____________.
`int_0^log5 (e^xsqrt(e^x - 1))/(e^x + 3)` dx = ______
Area enclosed between the curve y2(4 - x) = x3 and line x = 4 above X-axis is ______.
The area enclosed by the parabolas x = y2 - 1 and x = 1 - y2 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
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x, is:
The area of the circle `x^2 + y^2 = 16`, exterior to the parabola `y = 6x`
The area of the region bounded by the curve y = sin x and the x-axis in [–π, π] is ______.
Area in first quadrant bounded by y = 4x2, x = 0, y = 1 and y = 4 is ______.
If area of the region bounded by y ≥ cot( cot–1|In|e|x|) and x2 + y2 – 6 |x| – 6|y| + 9 ≤ 0, is λπ, then λ is ______.
Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.
