Advertisements
Advertisements
प्रश्न
Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y2 = x and the x-axis.
Advertisements
उत्तर
Solving x + y = 2 and y2 = x simultaneously, we get the points of intersection as (1, 1) and (4, –2).
The required area = the shaded area = `int_0^1 sqrt(x) dx + int_1^2 (2 - x) dx`
= `2/3 [x^(3/2)]_0^1 + [2x - x^2/2]_1^2`
= `2/3 + 1/2 = 7/6` suqare 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 parabola y2 = 4ax and its latus rectum.
Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.
Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2.
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.
Make a rough sketch of the graph of the function y = 4 − x2, 0 ≤ x ≤ 2 and determine the area enclosed by the curve, the x-axis and the lines x = 0 and 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.
Sketch the graph y = | x + 3 |. Evaluate \[\int\limits_{- 6}^0 \left| x + 3 \right| dx\]. What does this integral represent on the graph?
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 common to the parabolas 4y2 = 9x and 3x2 = 16y.
Using integration, find the area of the triangular region, the equations of whose sides are y = 2x + 1, y = 3x+ 1 and x = 4.
Find the area of the region bounded by \[y = \sqrt{x}, x = 2y + 3\] in the first quadrant and x-axis.
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.
Sketch the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 1. Also, find the area of this region.
Find the area of the region bounded by y = | x − 1 | and y = 1.
Find the area bounded by the lines y = 4x + 5, y = 5 − x and 4y = x + 5.
Find the area of the figure bounded by the curves y = | x − 1 | and y = 3 −| x |.
Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using horizontal strips.
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 curves y = sin x between the ordinates x = 0, x = π and the x-axis is _____________ .
The ratio of the areas between the curves y = cos x and y = cos 2x and x-axis from x = 0 to x = π/3 is ________ .
Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices area A(1, 2), B (2, 0) and C (4, 3).
Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x
Using the method of integration, find the area of the region bounded by the lines 3x − 2y + 1 = 0, 2x + 3y − 21 = 0 and x − 5y + 9 = 0
Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2
Find the area bounded by the curve y = sinx between x = 0 and x = 2π.
Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x2, 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.
Find the area of the region enclosed by the curves y2 = x, x = `1/4`, y = 0 and x = 1, using integration.
Make a rough sketch of the region {(x, y) : 0 ≤ y ≤ x2 + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} and find the area of the region, using the method of integration.
