# Chapter 9 - Multiveriable Calculus - Chapter Review - Review Exercises - Page 519: 69

$$\frac{2}{{135}}\left( {{{42}^{5/2}} - {{24}^{5/2}} - {{39}^{5/2}} + {{21}^{5/2}}} \right)$$

#### Work Step by Step

\eqalign{ & \int_3^4 {\int_2^5 {\sqrt {6x + 3y} dx} dy} \cr & = \int_3^4 {\left[ {\int_2^5 {\sqrt {6x + 3y} dx} } \right]dy} \cr & {\text{solve the inner integral}} \cr & \int_2^5 {\sqrt {6x + 3y} dx} = \frac{1}{6}\int_2^5 {\sqrt {6x + 3y} \left( 6 \right)dx} \cr & = \frac{1}{6}\left[ {\frac{{{{\left( {6x + 3y} \right)}^{3/2}}}}{{3/2}}} \right]_2^5 \cr & = \frac{1}{9}\left[ {{{\left( {6x + 3y} \right)}^{3/2}}} \right]_2^5 \cr & {\text{evaluate limits for }}x \cr & = \frac{1}{9}\left[ {{{\left( {6\left( 5 \right) + 3y} \right)}^{3/2}} - {{\left( {6\left( 2 \right) + 3y} \right)}^{3/2}}} \right] \cr & = \frac{1}{9}\left[ {{{\left( {30 + 3y} \right)}^{3/2}} - {{\left( {12 + 3y} \right)}^{3/2}}} \right] \cr & {\text{then}} \cr & \int_3^4 {\int_2^5 {\sqrt {6x + 3y} dx} dy} = \frac{1}{9}\int_3^4 {\left[ {{{\left( {30 + 3y} \right)}^{3/2}} - {{\left( {12 + 3y} \right)}^{3/2}}} \right]dy} \cr & = \frac{1}{{27}}\int_3^4 {\left[ {{{\left( {30 + 3y} \right)}^{3/2}}\left( 3 \right) - {{\left( {12 + 3y} \right)}^{3/2}}\left( 3 \right)} \right]dy} \cr & {\text{integrating by using the powe rule}} \cr & = \frac{1}{{27}}\left[ {\frac{{{{\left( {30 + 3y} \right)}^{5/2}}}}{{5/2}} - \frac{{{{\left( {12 + 3y} \right)}^{5/2}}}}{{5/2}}} \right]_3^4 \cr & = \frac{2}{{135}}\left[ {{{\left( {30 + 3y} \right)}^{5/2}} - {{\left( {12 + 3y} \right)}^{5/2}}} \right]_3^4 \cr & evaluating \cr & = \frac{2}{{135}}\left[ {{{\left( {30 + 3\left( 4 \right)} \right)}^{5/2}} - {{\left( {12 + 3\left( 4 \right)} \right)}^{5/2}}} \right] - \left[ {{{\left( {30 + 3\left( 3 \right)} \right)}^{5/2}} - {{\left( {12 + 3\left( 3 \right)} \right)}^{5/2}}} \right] \cr & = \frac{2}{{135}}\left[ {{{\left( {42} \right)}^{5/2}} - {{\left( {24} \right)}^{5/2}}} \right] - \left[ {{{\left( {39} \right)}^{5/2}} - {{\left( {21} \right)}^{5/2}}} \right] \cr & = \frac{2}{{135}}\left( {{{42}^{5/2}} - {{24}^{5/2}} - {{39}^{5/2}} + {{21}^{5/2}}} \right) \cr}

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