Chapter 8.4, Problem 39E

Find the area between the graph of the given function and the $x\text{-}$axis over the given interval, if possible.

Consider the functions $f\left(x\right)=1/\sqrt{1+x^2}\text{and}g\left(x\right)=1/\sqrt{1+x^4}$.

a. Use your calculator to approximate ${\displaystyle {\int }_1^{b}f\left(x\right)dx}$ for $b=20,50,100,1000,\text{ and }10,000$.

b. Based on your answer from part a, would you guess that ${\displaystyle {\int }_1^{\infty }f\left(x\right)dx}$ is convergent or divergent?

c. Use your calculator to approximate ${\displaystyle {\int }_1^{b}g\left(x\right)dx}$ for $b=20,50,100,1000,\text{ and }10,000$.

d. Based on your answer from part c, would you guess that ${\displaystyle {\int }_1^{\infty }g\left(x\right)dx}$ is convergent or divergent?

e. Show how the answer to parts b and d might be guessed by comparing the integrals with others whose convergence or divergence is known. (Hint: For large $x$, the difference between $1+x^2\text{}\text{and }{x}^2$ is relatively small.)

Note: The first integral is indeed divergent, and the second convergent, with an approximate value of $0.9270$.