Evaluate the following integral: ∫ x √(2 - x) dx, x ∈ [0, 2]

Question

Evaluate the following integral:

\(\int\limits_{0}^{2}\text x\sqrt{2-\text x}d\text x \)

Answer 1

Let I = \(\int\limits_{0}^{2}\text x\sqrt{2-\text x}d\text x \)...equation 1

Put 2 – x = y² 

Differentiating both sides

– dx = 2ydy

For x = 2

2 – x = y²

2 – 2 = y²

y = 0

For x = 0

2 – x = y²

2 – 0 = y²

y = √2

Substituting the values in equation 1

We know

b and a being the upper and lower limits respectively

thus