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Numerade Educator

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Problem 57 Medium Difficulty

Use the properties of integrals to verify the inequality without evaluating the integrals.

$ \displaystyle 2 \le \int^1_{-1} \sqrt{1 + x^2} \,dx \le 2\sqrt{2} $

Answer

$$2 \leq \int_{-1}^{1} \sqrt{1+x^{2}} d x \leq 2 \sqrt{2}$$

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Video Transcript

okay. We know the squirt of one has to be less than or equal to the square root of one plus X square, which is less than or equal to the sport of two. And the reason why is because zero is between is excellent exporters between zero and one because one is less than equal to one plus X squared, which is less than or equal to two. Therefore, we know we can write this as Tuas, less than or equal to the interval from negative one to the bounds of one times good of one plus X squared de axe, which is less than equal to two time squirt of two.