Advertisements
Advertisements
प्रश्न
Find the area of the smaller region bounded by the curves `x^2/25 + y^2/16` = 1 and `x/5 + y/4` = 1, using integration.
Advertisements
उत्तर
`x^2/25 + y^2/16` = 1 and `x/5 + y/4` = 1
`\implies` y = `4/5 sqrt(25 - x^2)`
The points of intersection of the given curve and line are A(0, 4) and B(5, 0).
Area of shaded region = Area of ellipse in I quadrant – Area of triangle ΔOAB
= `int_0^5 (4/5 sqrt(25 - x^2))dx - 1/2 xx 5 xx 4`
= `4/5 [x/2 sqrt(25 - x^2) + 25/2 sin^-1 x/5]_0^5 - 10 ...[∵ int (sqrt(a^2 - x^2))dx = x/2 sqrt(a^2 - x^2) + a^2/2 sin^-1 x/a]`
= `4/5 [0 + 25/2 sin^-1 1] - 10`
= `10 xx π/2 - 10`
= (5π – 10) sq. units.
APPEARS IN
संबंधित प्रश्न
Find the area bounded by the curve y2 = 4ax, x-axis and the lines x = 0 and x = a.
Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis
Prove that the curves y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.
Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.
Find the area of the region bounded by the parabola y2 = 4ax and the line x = a.
Draw a rough sketch of the curve y = \[\frac{\pi}{2} + 2 \sin^2 x\] and find the area between x-axis, the curve and the ordinates x = 0, x = π.
Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =\[\frac{\pi}{3}\] are in the ratio 2 : 3.
Compare the areas under the curves y = cos2 x and y = sin2 x between x = 0 and x = π.
Find the area of the region in the first quadrant bounded by the parabola y = 4x2 and the lines x = 0, y = 1 and y = 4.
Find the area of the region bounded by y =\[\sqrt{x}\] and y = x.
Using integration, find the area of the region bounded by the triangle whose vertices are (2, 1), (3, 4) and (5, 2).
Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x.
Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.
Find the area bounded by the curves x = y2 and x = 3 − 2y2.
Find the area of the region bounded by the curve y = \[\sqrt{1 - x^2}\], line y = x and the positive x-axis.
If the area bounded by the parabola \[y^2 = 4ax\] and the line y = mx is \[\frac{a^2}{12}\] sq. units, then using integration, find the value of m.
The area bounded by the curve y = loge x and x-axis and the straight line x = e is ___________ .
The area of the region formed by x2 + y2 − 6x − 4y + 12 ≤ 0, y ≤ x and x ≤ 5/2 is ______ .
The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .
Area lying in first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2, is
Using integration, find the area of the smaller region bounded by the ellipse `"x"^2/9+"y"^2/4=1`and the line `"x"/3+"y"/2=1.`
Find the area of the region bounded by the parabolas y2 = 6x and x2 = 6y.
The area enclosed by the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 is equal to ______.
Find the area of region bounded by the line x = 2 and the parabola y2 = 8x
The area of the region bounded by the circle x2 + y2 = 1 is ______.
Find the area of the region bounded by `x^2 = 4y, y = 2, y = 4`, and the `y`-axis in the first quadrant.
The area bounded by the curve `y = x|x|`, `x`-axis and the ordinate `x` = – 1 and `x` = 1 is given by
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 – 3x2 – 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______.
Using integration, find the area of the region bounded by line y = `sqrt(3)x`, the curve y = `sqrt(4 - x^2)` and Y-axis in first quadrant.
Using integration, find the area of the region bounded by the curve y2 = 4x and x2 = 4y.
