Solution of Chapter 21. Areas of Bounded Regions (RD Sharma - Mathematics (Volume 2) Book)

Using integration, find the area of the region bounded between the line x = 2 and the parabola y² = 8x.

Using integration, find the area of the region bounded by the line y – 1 = x, the x – axis and the ordinates x = – 2 and x = 3.

Find the area the region bounded by the parabola y² = 4ax and the line x = a.

Find the area lying above the x - axis and under the parabola y = 4x – x².

Draw a rough sketch to indicate the bounded between the curve y² = 4x and the line x = 3. Also, find the area of this region

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.

Sketch the graph of 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 above x - axis from x = 0 to x = 2. Draw a sketch of curve also.

Draw the rough sketch of y² + 1 = x, x ≤ 2. Find the area enclosed by the curve and the line x = 2

Draw a rough sketch of the graph of the curve and evaluate the area of the region under the curve and above the x - axis

Sketch the region {(x,y):9x² + 4y² = 36} and the find the area of the region enclosed by it, using integration

Draw a rough sketch of the graph of the function y=2√1–x²,x[0,1] and evaluate the are enclosed between the curve and the x–axis.

Determine the area under the included between the lines x = 0 and x = 1

Using integration, find the area of the region bounded by the line 2y = 5x + 7, x - axis the lines x = 2 and x = 8.

Using definite integrals, find the area of circle x² + y² = a²

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.

Sketch the graph y = |x - 5|. Evaluate . What does this value of the integral represent on the graph?

Sketch the graph y = |x + 3|. Evaluate . What does this integral represent on the graph?

Sketch the graph y = |x + 1|. Evaluate . What does the value of this integral represent on the graph?

Draw a rough sketch of the curve xy –3x – 2y – 10 = 0, x - axis and the lines x = 3, x = 4.

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.

Find the area bounded by the curve y = cosx, x - axis and the ordinates x = 0 and x = 2π.

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and are in the ration 2:3.

Compare the areas under the curves y = cos²x and y = sin² x between x = 0 and x = π.

Find the area bounded by the ellipse 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 x² + y² = a² cut off by the line .

Find the area of the region bounded by the curve x = at², y = 2at between the ordinates corresponding t = 1 and t = 2

Find the area enclosed by the curve x = 3 cost, y = 2 sint

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 bounded by x² = 16y, y = 1, y = 4 and the y – axis in the first quadrant.

Find the area of the region bounded by x² = 4ay and its latus rectum.

Find the area of the region bounded by x² + 16y = 0 and its latus rectum.

Find the area of the region bounded by the curve ay² = x³, they y-axis and the lines y = a and y = 2a.

Calculate the area of the region bounded by the parabolas y² = 6x and x² =6y.

Find the area of the region common to the parabolas 4y² = 9x and 3 x² =16y.

Find the area of the region bounded by y = √x and y = x

Find the area bounded by the curve y = 4 – x² and the lined y = 0, y = 3.

Find the area of the region .

Using integration find the area of the region bounded by the triangle whose vertices are (2,1), (3,4) and (5,2).

Using integration, find the area of the region bounded by the triangle ABC whose vertices A, B, C are ( – 1,1), (0,5) and (3,2) respectively.

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 {(x, y) : y²≤8x, x² + y²≤ 9}

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 circles x² + y² = 4 and (x – 2)² + y² = 4.

Find the area of the region included between the parabola y² = x and the line x + y = 2.

Draw a rough sketch of the region {(x, y) : y² ≤ 3x, 3x² + 3y² ≤ 16} and find the area enclosed by the region using method of integration.

Draw a rough sketch of the region {(x,y) : y² ≤ 5x, 5x² + 5y² ≤ 36} and find the area enclosed by the region using the method of integration.

Draw a rough sketch and find the area of the region bounded by the two parabolas y² = 4x and x² = 4y by using methods of integration.

Find the area included between the parabolasy² = 4ax and x² = 4by.

Prove that the area in the first quadrant enclosed by the axis, the line x = √3y and the circle x² + y² = 4 is π/3.

Find the area of the region bounded by by y = √x and 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 = 6ax.

OR

Find the area of the region {(x,y):y² ≤ 6ax} and {(x,y):x² + y² ≤ 16a²}.

Find the area, lying above x - axis and included between the circle circle x² + y² = 8x and the parabola y² = 4x.

Find the area enclosed by the parabolas y = 5x² and y = 2x² + 9.

Prove that the area common to the two parabolas y = 2x² and y = x² + 4 is 32/3 sq. Units.

Using integration, find the area of the region bounded by the triangle whose vertices are

(i) ( – 1, 2), (1, 5) and (3, 4)

(ii) ( – 2, 1), (0, 4) and (2, 3)

Find the area of the region bounded by y = √x and y = x.

Find the area of the region in the first quadrant enclosed by the x - axis, the line y = √3x and the circle x² + y² = 16.

Find the area of the region bounded by the parabola y² = 2x + 1 and the line x – y – 1 = 0.

Find the area of the region bounded by the curves y = x – 1 and (y – 1)² = 4 (x + 1).

Find the area enclosed by the curve y = – x² and the straight line x + y + 2 = 0

Find the area enclosed by the curve Y = 2 – x² and the straight line x + y = 0.

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

Sketch the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 1. Also, find the area of the region.

Find the area bounded by the curves x = y² and x = 3 – 2 y².

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 region {(x,y)|x – 1| ≤ y ≤ √5 – x²}.

Find the area of the region bounded by y – |x – 1| and y = 1.

Find the area of the region bounded by y = x and circle x² + y² = 32 in the 1^st quadrant.

Find the area of the circle x² + y² = 16 which is exterior the parabola y² = 6x.

Find the area of the region enclosed by the parabola x² = y and the line y = x + 2.

Make a sketch of the region{(x,y): 0 ≤ y ≤ x² + 3; 0 ≤ y ≤ 2x + 3; 0 ≤ x ≤ 3} and find its area using integration.

Find the area of the region bounded by the curve y = √1 – x², line y = x and the positive x - axis.

Find the area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5.

Find the area of the region enclosed between the two curves x² + y² = 9 and (x – 3)² + y² = 9.

Find the area of the region {(x,y): x² + y² ≤ 4, x + y ≥ 2}

Using integration, find the area of the following region.

Using integration find the area of the region bounded by the curve , x² + y² – 4x = 0 and the x-axis.

Find the area enclosed by the curves y = |x – 1| and y = – |x – 1| + 1.

Find the area enclosed by the curves 3x² + 5y = 32 and y = |x – 2|.

Find the area enclosed by the parabolas y = 4x – x² and y = x² – x.

In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x – x² and y = x² – x?

Find the area of the figure bounded by the curves y = |x – 1| and y = 3 – |x|.

If the area bounded by the parabola y² = 4ax and the line y = mx is a²/12 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 1024/3 square units, find the value of a.

Find the area of the region between the parabola x = 4y – y² and the line x = 2y – 3.

Find the area bounded by the parabola x = 8 + 2y – y²; the y - axis and the lines y = – 1 and y = 3.

Find the area bounded by the parabola y² = 4x and the line y = 2x – 4.

(i) By using horizontal strips

(ii) By using vertical strips.

Find the area of the region bounded by the parabola y² = 2x and the straight-line x – y = 4.

If the area above the x-axis, bounded by the curves y = 2^kx and x = 0, and x = 2 is 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_eand 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, latus rectum and x-axis is

The area of the region is

The area common to 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 = cosx 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 ≤ 2π 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 ≤ π/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 cure y² = 4x, y-axis and the line y = 3, is