Advertisements
Advertisements
Question
Sketch the region `{(x, 0) : y = sqrt(4 - x^2)}` and x-axis. Find the area of the region using integration.
Advertisements
Solution
Given that `{(x, 0) : y = sqrt(4 - x^2)}`
⇒ y2 = 4 – x2
⇒ x2 + y2 = 4 which is a circle.
Required area = `2 * int_0^2 sqrt(4 - x^2) "d"x`
Since circle is symmetrical about y-axis
= `2 * int_0^2 sqrt((2)^2 - x^2) "d"x`
= `2 * [x/2 sqrt(4 - x^2) + 4/2 sin^-1 x/2]_0^2`
= `2[(2/2 sqrt(4 - 4) + 2 sin^-1 (1)) - (0 + 0)]`
= `2[2 * pi/2]`
= 2π sq.units
Hence, the required area = 2π sq.units
APPEARS IN
RELATED QUESTIONS
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum.
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.
Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
The area bounded by the curve y = x | x|, x-axis and the ordinates x = –1 and x = 1 is given by ______.
[Hint: y = x2 if x > 0 and y = –x2 if x < 0]
Find the area of ellipse `x^2/1 + y^2/4 = 1`
Draw a rough sketch of the curve and find the area of the region bounded by curve y2 = 8x and the line x =2.
Using integration, find the area of the region bounded between the line x = 2 and the parabola y2 = 8x.
Find the area lying above the x-axis and under the parabola y = 4x − x2.
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.
Sketch the region {(x, y) : 9x2 + 4y2 = 36} and find the area of the region enclosed by it, using integration.
Using integration, find the area of the region bounded by the line 2y = 5x + 7, x-axis and the lines x = 2 and x = 8.
Sketch the graph y = | x − 5 |. Evaluate \[\int\limits_0^1 \left| x - 5 \right| dx\]. What does this value of the integral represent on the graph.
Sketch the graph y = |x + 1|. Evaluate\[\int\limits_{- 4}^2 \left| x + 1 \right| dx\]. What does the value of this integral represent on the graph?
Find the area of the region bounded by the curve xy − 3x − 2y − 10 = 0, x-axis and the lines x = 3, x = 4.
Compare the areas under the curves y = cos2 x and y = sin2 x between x = 0 and x = π.
Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.
Using integration, find the area of the region bounded by the triangle whose vertices are (2, 1), (3, 4) and (5, 2).
Prove that the area in the first quadrant enclosed by the x-axis, the line x = \[\sqrt{3}y\] and the circle x2 + y2 = 4 is π/3.
Find the area of the region in the first quadrant enclosed by x-axis, the line y = \[\sqrt{3}x\] and the circle x2 + y2 = 16.
Find the area bounded by the parabola y = 2 − x2 and the straight line y + x = 0.
Using integration, find the area of the triangle ABC coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).
Using integration, find the area of the following region: \[\left\{ \left( x, y \right) : \frac{x^2}{9} + \frac{y^2}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2} \right\}\]
Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using horizontal strips.
If An be the area bounded by the curve y = (tan x)n and the lines x = 0, y = 0 and x = π/4, then for x > 2
The area bounded by the curves y = sin x between the ordinates x = 0, x = π and the x-axis is _____________ .
The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .
The area of the circle x2 + y2 = 16 enterior to the parabola y2 = 6x is
Area lying in first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2, is
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2
Using integration, find the area of the region bounded by the line 2y = 5x + 7, x- axis and the lines x = 2 and x = 8.
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Find the area bounded by the curve y = sinx between x = 0 and x = 2π.
Compute the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.
Using integration, find the area of the region bounded between the line x = 4 and the parabola y2 = 16x.
If a and c are positive real numbers and the ellipse `x^2/(4c^2) + y^2/c^2` = 1 has four distinct points in common with the circle `x^2 + y^2 = 9a^2`, then
Find the area of the region bounded by `x^2 = 4y, y = 2, y = 4`, and the `y`-axis in the first quadrant.
Let the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) is equal to ______.
The area of the region S = {(x, y): 3x2 ≤ 4y ≤ 6x + 24} is ______.
The area (in square units) of the region bounded by the curves y + 2x2 = 0 and y + 3x2 = 1, is equal to ______.
