Question

Roundabouts are often made on busy roads to ease the traffic and avoid red lights.

One such round-about is made such that equation representing its boundary is given by C₁ ;  x² + y² = 64.

There is a circular pond with a fountain in the middle of the roundabout whose equation is given by C₂ ;  x² + y² = 64.

Based on the given information, answer the following questions:

1. Represent the given equations C₁ and C₂ with the help of a diagram.
2. Express y as a function of x, (y = f(x)), for both C₁ and C₂.
3. 1. Using integration find the area of region covered by the roundabout.
      OR
   2. Using integration, find the area of region covered by circular pond.

Answer 1

(i)

The equations represent two concentric circles centered at the origin (0, 0):

• C₁ (Roundabout): x² + y² = 8² (Radius = 8 units)
• C₂ (Circular Pond): x² + y² = 2² (Radius = 2 units)

(ii)

To express y = f(x), we isolate y in both equations:

• For C₁: y = `+-sqrt(64 - x^2)`
• For C₂: y = `+-sqrt(4 - x^2)`

(iii) (a)

To find the area of the entire roundabout (C₁), we calculate the area in the first quadrant and multiply by 4:

Area = `4 int_0^8 sqrt(8^2 - x^2)  dx`

Using the identity `int sqrt(a^2 - x^2)  dx = x/2 sqrt(a^2 - x^2) + a^2/2 sin^-1 (x/a)`:

Area = `4[x/2 sqrt(64 - x^2) + 64/2 sin^-1 (x/8)]_0^8`

Area = `4[(0 + 32 * π/2) - (0 + 0)]`

Area = 4(16π)

∴ Area = 64π sq. units

OR

(iii) (b)

Similarly, for the pond (C₂) with radius r = 2:

Area = `4 int_0^2 sqrt(2^2 - x^2)  dx`

Area = `4[ x/2 sqrt(4 - x^2) + 4/2 sin^-1 (x/2)]_0^2`

Area = `4[(0 + 2 * π/2) - (0 + 0)]`

Area = 4(π)

∴ Area = 4π sq. units