RD Sharma solutions for Mathematics [English] Class 12 chapter 21 - Areas of Bounded Regions [Latest edition]

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.

Find the area under the curve y = \[\sqrt{6x + 4}\] above x-axis from x = 0 to x = 2. Draw a sketch of curve also.

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.

Draw a rough sketch of the graph of the function y = 2 \[\sqrt{1 - x^2}\] , x ∈ [0, 1] and evaluate the area enclosed between the curve and the x-axis.

Determine the area under the curve y = \[\sqrt{a^2 - x^2}\]  included between the lines x = 0 and x = a.

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 + 3 |. Evaluate \[\int\limits_{- 6}^0 \left| x + 3 \right| dx\]. What does this 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?

Draw a rough sketch of the curve y = \[\frac{\pi}{2} + 2 \sin^2 x\] and find the area between x-axis, the curve and the ordinates x = 0, x = π.

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.

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =\[\frac{\pi}{3}\]  are in the ratio 2 : 3.

Find the area bounded by the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]  and the ordinates x = ae and x = 0, where b² = a² (1 − e²) and e < 1.

Find the area of the minor segment of the circle \[x^2 + y^2 = a^2\] cut off by the line \[x = \frac{a}{2}\]

Find the area of the region bounded by the curve \[x = a t^2 , y = 2\text{ at }\]between the ordinates corresponding t = 1 and t = 2.

Find the area enclosed by the curve x = 3cost, y = 2sin t.

Find the area of the region bounded by the curve \[a y^2 = x^3\], the y-axis and the lines y = a and y = 2a.

Find the area of the region bounded by y =\[\sqrt{x}\] and y = x.

Find the area of the region \[\left\{ \left( x, y \right): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \leq \frac{x}{a} + \frac{y}{b} \right\}\]

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 bounded by \[y = \sqrt{x}, x = 2y + 3\]  in the first quadrant and x-axis.

Find the area common to the circle x² + y² = 16 a² and the parabola y² = 6 ax.
                                   OR
Find the area of the region {(x, y) : y² ≤ 6ax} and {(x, y) : x² + y² ≤ 16a²}.

Prove that the area common to the two parabolas y = 2x² and y = x² + 4 is \[\frac{32}{3}\] sq. units.

Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.

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 enclosed by the curve \[y = - x^2\] and the straight line x + y + 2 = 0. 

Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.

Using integration find the area of the region:
\[\left\{ \left( x, y \right) : \left| x - 1 \right| \leq y \leq \sqrt{5 - x^2} \right\}\]

Find the area of the region bounded by the curve y = \[\sqrt{1 - x^2}\], line y = x and the positive x-axis.

Using integration, find the area of the following region: \[\left\{ \left( x, y \right) : \frac{x^2}{9} + \frac{y^2}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2} \right\}\]

Using integration find the area of the region bounded by the curves \[y = \sqrt{4 - x^2}, x^2 + y^2 - 4x = 0\] and the x-axis.

If the area bounded by the parabola \[y^2 = 4ax\] and the line y = mx is \[\frac{a^2}{12}\] sq. units, then using integration, find the value of m. 

If the area enclosed by the parabolas y² = 16ax and x² = 16ay, a > 0 is \[\frac{1024}{3}\] square units, find the value of a.

Find the area bounded by the parabola y² = 4x and the line y = 2x − 4 By using horizontal strips.

Find the area bounded by the parabola y² = 4x and the line y = 2x − 4 By using vertical strips.

If the area above the x-axis, bounded by the curves y = 2^kx and x = 0, and x = 2 is \[\frac{3}{\log_e 2}\], then the value of k is __________ .

The area included between the parabolas y² = 4x and x² = 4y is (in square units)

The area bounded by the curve y = log_e x and x-axis and the straight line x = e is ___________ .

The area bounded by y = 2 − x² and x + y = 0 is _________ .

The area bounded by the parabola x = 4 − y² and y-axis, in square units, is ____________ .

If A_n 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 of the region formed by x² + y² − 6x − 4y + 12 ≤ 0, y ≤ x and x ≤ 5/2 is ______ .

The area enclosed between the curves y = log_e (x + e), x = log_e \[\left( \frac{1}{y} \right)\] and the x-axis is _______ .

The area of the region bounded by the parabola (y − 2)² = x − 1, the tangent to it at the point with the ordinate 3 and the x-axis is _________ .

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 and x² = 4ay is ___________ .

The area bounded by the curve y = x⁴ − 2x³ + x² + 3 with x-axis and ordinates corresponding to the minima of y is _________ .

The area bounded by the parabola y² = 4ax, latusrectum and x-axis is ___________ .

The area of the region \[\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}\] is __________ .

The closed area made by the parabola y = 2x² and y = x² + 4 is __________ .

The area of the region bounded by the parabola y = x² + 1 and the straight line x + y = 3 is given by

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 ________ .

The area between x-axis and curve y = cos x when 0 ≤ x ≤ 2 π is ___________ .

Area bounded by parabola y² = x and straight line 2y = x is _________ .

The area bounded by the curve y = 4x − x² and the x-axis is __________ .

Area enclosed between the curve y² (2a − x) = x³ and the line x = 2a above x-axis is ___________ .

The area of the region (in square units) bounded by the curve x² = 4y, line x = 2 and x-axis is

The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is (b −1) sin (3b + 4). Then, f (x) is __________ .

The area bounded by the curve y² = 8x and x² = 8y is ___________ .

The area bounded by the parabola y² = 8x, the x-axis and the latusrectum is ___________ .

Area bounded by the curve y = x³, the x-axis and the ordinates x = −2 and x = 1 is ______.

The area bounded by the curve y = x |x| and the ordinates x = −1 and x = 1 is given by

The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \[\frac{\pi}{2}\] is _________ .

The area of the circle x² + y² = 16 enterior to the parabola y² = 6x is

Smaller area enclosed by the circle x² + y² = 4 and the line x + y = 2 is

Area lying between the curves y² = 4x and y = 2x is

Area lying in first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2, is

Area of the region bounded by the curve y² = 4x, y-axis and the line y = 3, is