Maths · Determining areas of the regions bounded by simple curves in standard forms
The area bounded by the axis, the curve \) and the lines and is equal to \) for
The area bounded by the \( x- \) axis, the curve \( y=f(x) \) and the lines \( x=1 \) and \( x=b \) is equal to \( (\sqrt{b^{2}+1}-\sqrt{2}) \) for all \( \boldsymbol{b}>1, \) then \( \boldsymbol{f}(\boldsymbol{x}) \) is
- A. \( \sqrt{x-1} \)
- B. \( \sqrt{x+1} \)
- C. \( \sqrt{x^{2}+1} \)
- D. \( \frac{x}{\sqrt{x^{2}+1}} \)
Step-by-step solution
The area from x=1 to x=b is ∫₁ᵇ f(x) dx = √(b²+1) - √2. Differentiating both sides w.r.t. b gives f(b) = b/√(b²+1), so f(x) = x/√(x²+1).
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