What is \int_{\sqrt{2}}^{\sqrt{3}}f(x)\,dx equal to?
Consider the following for the two (02) items that follow : Let f(x)=[x^{2}] where [~] is the greatest integer function.
- A. \sqrt{3}-\sqrt{2}
- B. 2(\sqrt{3}-\sqrt{2}) ✓
- C. 3-\sqrt{2}
- D. 1
Correct Answer: B. 2(\sqrt{3}-\sqrt{2})
Explanation
In the interval (\sqrt{2}, \sqrt{3}), the value of x^2 lies strictly between 2 and 3. Consequently, the greatest integer [x^2] = 2. The integral becomes \int_{\sqrt{2}}^{\sqrt{3}} 2 \, dx = 2[x]_{\sqrt{2}}^{\sqrt{3}} = 2(\sqrt{3}-\sqrt{2}).
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