Area Under The Curves

The area bounded by the curves \(y=\left|x^{2}-1\right|\) and \(y=1\) is

The area of the region enclosed between the parabolas y2 = 2x \(-\) 1 and y2 = 4x \(-\) 3 is

Let the locus of the centre \((\alpha, \beta), \beta>0\), of the circle which touches the circle \(x^{2}+(y-1)^{2}=1\) externally and also touches the \(x\)-axis be \(\mathrm{L}\). Then the area bounded by \(\mathrm{L}\) and the line $$y=4$...

The area of the region given by
\(A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}\) is :

Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve \(4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0\) at the point (\(-\)2, 3) be A. Then 8A is equal to ______________.

Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then \({{{R_2}} \over {{R_1}}}\) is equal to ______________.

The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________.

If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = \({3 \over 2}\) and the curve y = 1 + 4x \(-\) x2, then 12 m is equal to _____________.

If the area of the bounded region \(R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}\) is , \(\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma\), then the value of $${(\alpha + \beta...

The area of the region bounded by y \(-\) x = 2 and x2 = y is equal to :

The area of the region bounded by the parabola (y \(-\) 2)2 = (x \(-\) 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

The area bounded by the lines y = || x \(-\) 1 | \(-\) 2 | is ___________.

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = \({\pi \over 2}\) in the first quad...

The area of the region \(S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\}\) is ____________.

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 t...

The area (in sq. units) of the region, given by the set \(\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\}\) is :

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to __________.

The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola
y2 = 9x, is :

The area of the region : \(R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\}\) is :

The area (in sq. units) of the region bounded by the curves x2 + 2y \(-\) 1 = 0, y2 + 4x \(-\) 4 = 0 and y2 \(-\) 4x \(-\) 4 = 0, in the upper half plane is _______________.

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 \(\alpha\)\(\sqrt 5\) + \(\beta\) + \(\gamma\) cos\(-\)1$$\left( ...

The area, enclosed by the curves \(y = \sin x + \cos x\) and \(y = \left| {\cos x - \sin x} \right|\) and the lines \(x = 0,x = {\pi \over 2}\), is :

The area bounded by the curve 4y2 = x2(4 \(-\) x)(x \(-\) 2) is equal to :

Let f : [\(-\)3, 1] \(\to\) R be given as \(f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.\)If the area bounded by y = f(x) and x-axis i...

Given : \(f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr {1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr } } \right.\)
and $$g(x) = \left( {x - {1 \over 2}}...

For a > 0, let the curves C1 : y2 = ax and
C2 : x2 = ay intersect at origin O and a point P.
Let the line x = b (0 < b < a) intersect the chord
OP and the x-axis at points Q and R,
respectively. If the line x = b bisects the area
bounded by...

The area (in sq. units) of the region
{(x,y) \(\in\) R2 : x2 \(\le\) y \(\le\) 3 – 2x}, is :

The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x, is:

The area (in sq. units) of the region
{(x, y) \(\in\) R2 | 4x2 \(\le\) y \(\le\) 8x + 12} is :

The area (in sq. units) of the region A = {(x, y) : |x| + |y| \(\le\) 1, 2y2 \(\ge\) |x|}

The area (in sq. units) of the region enclosed
by the curves y = x2 – 1 and y = 1 – x2 is equal to :

The area (in sq. units) of the region
A = {(x, y) : (x – 1)[x] \(\le\) y \(\le\) 2\(\sqrt x\), 0 \(\le\) x \(\le\) 2}, where [t]
denotes the greatest integer function, is :

The area (in sq. units) of the region
{ (x, y) : 0 \(\le\) y \(\le\) x2 + 1, 0 \(\le\) y \(\le\) x + 1,
\({1 \over 2}\) \(\le\) x \(\le\) 2 } is :

Area (in sq. units) of the region outside
\({{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1\) and inside the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 9} = 1\) is :

Consider a region R = {(x, y) \(\in\) R : x2 \(\le\) y \(\le\) 2x}.
if a line y = \(\alpha\) divides the area of region R into
two equal parts, then which of the following is
true?

The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is :

The area of the region
A = {(x, y) : 0 \(\le\) y \(\le\)x |x| + 1  and  \(-\)1 \(\le\) x \(\le\)1} in sq. units, is :

The area (in sq. units) of the region
A = {(x, y) : x2 \(\le\) y \(\le\) x + 2} is

The area (in sq. units) of the region
A = {(x, y) : \({{y{}^2} \over 2}\) \(\le\) x \(\le\) y + 4} is :-

The area (in sq. units) of the region
A = { (x, y) \(\in\) R × R|  0 \(\le\) x \(\le\) 3, 0 \(\le\) y \(\le\) 4,
y \(\le\) x2 + 3x} is :

Let S(\(\alpha\)) = {(x, y) : y2
\(\le\) x, 0 \(\le\) x \(\le\) \(\alpha\)} and A(\(\alpha\))
is area of the region S(\(\alpha\)). If for a \(\lambda\), 0 < \(\lambda\) < 4,
A(\(\lambda\)) : A(4) = 2 : 5, then \(\lambda\) e...

The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is

If the area (in sq. units) of the region {(x, y) : y2
\(\le\) 4x, x + y \(\le\) 1, x \(\ge\) 0, y \(\ge\) 0} is a \(\sqrt 2\) + b, then a – b is equal
to :

If the area (in sq. units) bounded by the parabola y2
= 4\(\lambda\)x and the line y = \(\lambda\)x, \(\lambda\) > 0, is \({1 \over 9}\)
, then \(\lambda\) is equal to :

The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is :

The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is :

If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is -

The area (in sq.units) of the region bounded by the curves y = 2x
and y = |x + 1|, in the first quadrant is :

If the area of the region bounded by the curves, \(y = {x^2},y = {1 \over x}\) and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal to :

The area (in sq. units) of the region
{x \(\in\) R : x \(\ge\) 0, y \(\ge\) 0, y \(\ge\) x \(-\) 2  and y \(\le\) \(\sqrt x\)}, is :