Question

The area bounded by the X-axis, the curve y = f(x) and the lines x = 1, x = b is equal to `sqrt("b"^2 + 1) - sqrt(2)` for all b > 1, then f(x) is ______.

Options

• `sqrt(x - 1)`

• `sqrt(x + 1)`

• `sqrt(x^2 + 1)`

• `x/sqrt(1 + x^2)`

Answer 1

The area bounded by the X-axis, the curve y = f(x) and the lines x = 1, x = b is equal to `sqrt("b"^2 + 1) - sqrt(2)` for all b > 1, then f(x) is `x/sqrt(1 + x^2)`.

Explanation:

According to the given condition,

`int_1^"b" "f"(x)  "d"x = sqrt("b"^2 + 1) - sqrt(2)`

= `sqrt("b"^2 + 1) - sqrt(1 + 1)`

= `[sqrt(x^2 + 1)]_1^"b"`

⇒ f(x) = `x/sqrt(1 + x^2)`