Determining If the Area Under a Graph Is Defined

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.