Chapter 8: Techniques of Integration - Practice Exercises - Page 519: 55

$$6$$

$$\eqalign{ & \int_0^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} \cr & {\text{The function }}f\left( y \right) = \frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}{\text{ is discontinuous at }}y = 1,\cr &{\text{so the integral is not defined for y=1}}. \cr &{\text{By the definition of the improper integrals, we have}} \cr & \int_0^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = \mathop {\lim }\limits_{b \to {1^ - }} \int_0^b {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} + \mathop {\lim }\limits_{a \to {0^ + }} \int_a^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} \cr & \cr & {\text{Integrating }}\int {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} \cr & \,\,\,\,\,\,\int {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = \int {{{\left( {y - 1} \right)}^{ - 2/3}}dy} . \cr & {\text{Use the power rule for integrals}} \cr & \,\,\,\,\,\,\, = \frac{{{{\left( {y - 1} \right)}^{1/3}}}}{{1/3}} + C \cr & \,\,\,\,\,\,\, = 3\root 3 \of {y - 1} + C \cr & \cr & {\text{Then}}{\text{,}} \cr & \cr & \mathop {\lim }\limits_{b \to {1^ - }} \int_0^b {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = \mathop {\lim }\limits_{b \to {1^ - }} \left( {3\root 3 \of {y - 1} } \right)_0^b \cr & \, = \mathop {\lim }\limits_{b \to {1^ - }} \left( {3\root 3 \of {b - 1} - 3\root 3 \of {0 - 1} } \right) \cr & \, = \mathop {\lim }\limits_{b \to {1^ - }} \left( {3\root 3 \of {b - 1} + 3} \right) \cr & {\text{Evaluate the limit when }}b \to {1^ - } \cr & \, = \left( {3\root 3 \of {1 - 1} + 3} \right) \cr & \, = 3 \cr & \mathop {\lim }\limits_{b \to {1^ - }} \int_0^b {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = 3 \cr & \cr & and \cr & \cr & \mathop {\lim }\limits_{a \to {0^ + }} \int_a^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = \mathop {\lim }\limits_{a \to {0^ + }} \left( {3\root 3 \of {y - 1} } \right)_a^2 \cr & \, = \mathop {\lim }\limits_{a \to {0^ + }} \left( {3\root 3 \of {2 - 1} - 3\root 3 \of {a - 1} } \right) \cr & \, = \mathop {\lim }\limits_{a \to {0^ + }} \left( {3 - 3\root 3 \of {a - 1} } \right) \cr & {\text{Evaluate the limit when }}a \to {0^ + } \cr & \, = \left( {3 - 3\root 3 \of {1 - 1} } \right) \cr & \, = 3 + 0 \cr & \mathop {\lim }\limits_{a \to {0^ + }} \int_a^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = 3 \cr & \cr & {\text{Since both of these integrals converge}},{\text{ the improper integral is}} \cr & \int_0^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = 3 + 3 \cr & \int_0^2 {\frac{{dy}}{{{{\left( {y - 1} \right)}^{2/3}}}}} = 6 \cr} $$