?(a) If \(g(x)=\left(\sin ^{2} x\right) / x^{2}\), use a calculator or computer to make

Chapter 7, Problem 55

(choose chapter or problem)

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

Equation Transcription:

g(x) = (sin2x)/x2

∫g(x) dx

t = 2, 5, 10, 100, 1000, and 10,000

∫g(x) dx

f(x) = 1/x2

∫g(x) dx

f

g

1 ⩽ x ⩽ 10

∫g(x) dx

Text Transcription:

g(x) = (sin^2x)/x^2

integral _1 ^t g(x) dx

t = 2, 5, 10, 100, 1000, and 10,000

integral _1 ^infinity g(x) dx

f(x) = 1/x^2

integral _1 ^infinity g(x) dx

f

g

1 leqslant x leqslant 10

integral _1 ^infinity g(x) dx