Mathematics

Problem

Consider the function

q(x)=\begin{cases} \dfrac{1}{(x-1)^{2}}, & 0\le x<1,\\ 2-x, & 1< x\le 2. \end{cases}

Which of the following statements is true concerning the area between the graph of

q

and the

x

–axis on

0\le x\le 2

? a) The area is finite and equals

\displaystyle\int_{0}^{1}\frac{1}{(x-1)^{2}}\,dx+\int_{1}^{2}(2-x)\,dx

. b) The area is infinite because both

\displaystyle\int_{0}^{1}\frac{1}{(x-1)^{2}}\,dx

and

\displaystyle\int_{1}^{2}(2-x)\,dx

diverge. c) The area is infinite because

\displaystyle\int_{0}^{1}\frac{1}{(x-1)^{2}}\,dx

diverges, whereas

\displaystyle\int_{1}^{2}(2-x)\,dx

is finite. d) The area is finite and equals

\dfrac{3}{2}