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WZ
Numerade Educator

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Problem 47 Hard Difficulty

(a) If $ g(x) = \frac{(\sin^2 x)}{x^2} $, use your calculator or computer to make a table of approximate values of $ \displaystyle \int_1^t g(x)\ dx $ for $ t = 2, 5, 10, 100, 1000 $, and $ 10,000 $. Does it appear that $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent?
(b) Use the Comparison Theorem with $ f(x) = \frac{1}{x^2} $ to show that $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent.
(c) Illustrate part (b) by graphing $ f $ and $ g $ on the same screen for $ 1 \le x \le 10 $. Use your graph to explain intuitively why $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent.

Answer

a. $g(x)=\frac{\sin ^{2} x}{x^{2}}$
b. $\int_{1}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x$ is convergent
c. convergent

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Video Transcript

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