Find the area (in sq. units) of the region bounded by 𝑦2=𝑥 and 𝑦2=2−𝑥

To find the area bounded by the curves 𝑦²=𝑥 and 𝑦²=2−𝑥, we first need to find the points of intersection.

Step 1: Set 𝑦²=𝑥 and 𝑦²=2−𝑥 equal to each other to find the x-values of the points of intersection.

x = 2 - x 2x = 2 x = 1

Step 2: Substitute x = 1 into 𝑦²=𝑥 and 𝑦²=2−𝑥 to find the corresponding y-values.

y² = 1 => y = ±1

So, the points of intersection are (1,1) and (1,-1).

Step 3: Set up the integral to find the area. The area A of the region bounded by the curves is given by the integral of the absolute difference of the functions over the interval from x = 0 to x = 1.

A = ∫ from 0 to 1 (sqrt(x) - sqrt(2-x)) dx + ∫ from 0 to 1 (-sqrt(x) + sqrt(2-x)) dx

Step 4: Evaluate the integral.

A = [2/3 x^(3/2) - 2/3 (2-x)^(3/2)] from 0 to 1 + [-2/3 x^(3/2) + 2/3 (2-x)^(3/2)] from 0 to 1 A = 2/3 + 2/3 = 4/3 sq. units

So, the area of the region bounded by 𝑦²=𝑥 and 𝑦²=2−𝑥 is 4/3 square units.