Determine if the area under the graph of $y=\frac{1}{x^{2}}$ to the right of $x=1$ is defined.

Solution See Figure 22. To determine if the area is defined, we examine $\int_{1}^{\infty }\,\frac{1}{x^{2}}dx$. $$\int_{1}^{\infty }\frac{1}{x^{2}}dx\hbox{:}\quad \lim_{b\,\rightarrow \,\infty }\int_{1}^{b}\frac{1}{x^{2}}dx= \lim_{b\,\rightarrow \,\infty } \left[ -\frac{1}{x}\right] _{1}^{b}= \lim_{b\,\rightarrow \,\infty }\left( - \frac{1}{b}+1\right) =1$$

The area under the graph $f(x)= \frac{1}{x^{2}}$ to the right of $1$ is defined and equals $1$ square unit.