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.
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
Sketch the region bounded by the curves `y=sqrt(5-x^2)` and y=|x-1| and find its area using integration.
Find the area of the region bounded by the curve x2 = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.
Area bounded by the curve y = x3, the x-axis and the ordinates x = –2 and x = 1 is ______.
Using integration, find the area of the region bounded by the line y − 1 = x, the x − axis and the ordinates x= −2 and x = 3.
Sketch the graph of y = \[\sqrt{x + 1}\] in [0, 4] and determine the area of the region enclosed by the curve, the x-axis and the lines x = 0, x = 4.
Draw a rough sketch of the graph of the curve \[\frac{x^2}{4} + \frac{y^2}{9} = 1\] and evaluate the area of the region under the curve and above the x-axis.
Find the area of the region bounded by x2 = 4ay and its latusrectum.
Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.
Find the area of the region bounded by y =\[\sqrt{x}\] and y = x.
Find the area bounded by the curve y = 4 − x2 and the lines y = 0, y = 3.
Find the area of the region bounded by the parabola y2 = 2x + 1 and the line x − y − 1 = 0.
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2.
In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x − x2 and y = x2− x?
Find the area bounded by the parabola x = 8 + 2y − y2; the y-axis and the lines y = −1 and y = 3.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x − y = 4.
The area bounded by the curve y = x |x| and the ordinates x = −1 and x = 1 is given by
Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3, is
The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is ______.
Find the area bounded by the curve y = sinx between x = 0 and x = 2π.
Find the area of the region bounded by `x^2 = 4y, y = 2, y = 4`, and the `y`-axis in the first quadrant.
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9` = 1.
Smaller area bounded by the circle `x^2 + y^2 = 4` and the line `x + y = 2` is.
Area (in sq.units) of the region outside `|x|/2 + |y|/3` = 1 and inside the ellipse `x^2/4 + y^2/9` = 1 is ______.
Let f : [–2, 3] `rightarrow` [0, ∞) be a continuous function such that f(1 – x) = f(x) for all x ∈ [–2, 3]. If R1 is the numerical value of the area of the region bounded by y = f(x), x = –2, x = 3 and the axis of x and R2 = `int_-2^3 xf(x)dx`, then ______.
Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the Y-axis. Hence, obtain its area using integration.
Using integration, find the area of the region bounded by the curve y2 = 4x and x2 = 4y.
