Advertisements
Advertisements
Question
Find the area of the region bounded by the curve ay2 = x3, the y-axis and the lines y = a and y = 2a.
Solution
We have
Area BMNC = `int_"a"^(2"a") x"d"y`
= `int_"a"^(2"a") "a"^(1/3) y^(2/3) "d"y`
= `(3"a"^(1/3))/5|y^(5/3)|_"a"^(2"a")`
= `(3"a"^(1/3))/5|(2"a")^(5/3) - "a"^(5/3)|`
= `3/5 "a"^(1/3) "a"^(5/3) |(2)^(5/3) - 1|`
= `3/5 "a"^2 |2.2^(2/3) - 1|` sq.units
APPEARS IN
RELATED QUESTIONS
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum.
triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.
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.
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 following curves, after making a rough sketch: y = 1 + | x + 1 |, x = −2, x = 3, y = 0.
Draw a rough sketch of the curve \[y = \frac{x}{\pi} + 2 \sin^2 x\] and find the area between the x-axis, the curve and the ordinates 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 common to the circle x2 + y2 = 16 and the parabola y2 = 6x.
Find the area of the region between the circles x2 + y2 = 4 and (x − 2)2 + y2 = 4.
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.
The area bounded by the parabola x = 4 − y2 and y-axis, in square units, 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
Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of the latus rectum is 10. Also, find its eccentricity.
Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x
Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.
Find the area of the region bounded by the curve y2 = 4x, x2 = 4y.
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.
The area of the region bounded by the curve y = `sqrt(16 - x^2)` and x-axis is ______.
Let f(x) be a continuous function such that the area bounded by the curve y = f(x), x-axis and the lines x = 0 and x = a is `a^2/2 + a/2 sin a + pi/2 cos a`, then `f(pi/2)` =
Area of the region bounded by the curve `y^2 = 4x`, `y`-axis and the line `y` = 3 is:
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9` = 1.
The area bounded by the curve `y = x|x|`, `x`-axis and the ordinate `x` = – 1 and `x` = 1 is given by
Let T be the tangent to the ellipse E: x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 and x = `sqrt(5)` is `sqrt(5)`α + β + γ `cos^-1(1/sqrt(5))`, then |α + β + γ| is equal to ______.
The area (in square units) of the region bounded by the curves y + 2x2 = 0 and y + 3x2 = 1, is equal to ______.
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.
