Question

If f(x) = `{{:(x^2",", "where"  0 ≤ x < 1),(sqrt(x)",", "when"  1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.

Options

• `1/3(4sqrt(2) - 1)`

• `1/3(4sqrt(2) + 1)`

• 0

• does not exist

Answer 1

If f(x) = `{{:(x^2",", "where"  0 ≤ x < 1),(sqrt(x)",", "when"  1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals `underlinebb(1/3(4sqrt(2) - 1))`.

Explanation:

I = `int_0^1x^2dx + int_1^2sqrt(x)dx`

= `[x^3/3]_0^1 + 2/3[x^(3/2)]_1^2`

= `1/3 + 2/3(2^(3/2) - 1)`

= `(1 + 4sqrt(2) - 2)/3`

= `1/3(4sqrt(2) - 1)`