The integral \(\mathop \smallint \limits_0^1 \mathop \smallint \l
A. <span class="math-tex">\(\frac{{26}}{{105}}\)</span>
B. <span class="math-tex">\(\frac{4}{{105}}\)</span>
C. <span class="math-tex">\(\frac{{12}}{{105}}\)</span>
D. <span class="math-tex">\(\frac{{16}}{{105}}\)</span>
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Right Answer is: A
SOLUTION
\(f\left( x \right) = \mathop \smallint \limits_0^1 \mathop \smallint \limits_0^{{x^2}} \left( {{x^2} + {y^2}} \right)dxdy\)
\(= \mathop \smallint \limits_0^1 \mathop \smallint \limits_0^{{x^2}} \left( {{x^2} + {y^2}} \right) \cdot dy \cdot dx\)
\(= \mathop \smallint \limits_0^1 \left[ {{x^2}y + \frac{{{y^3}}}{3}} \right]_0^{{x^2}}dx\)
\(= \mathop \smallint \limits_0^1 \left[ {{x^4} + \frac{{{x^6}}}{3}} \right]dx\)
\(= \left[ {\frac{{{x^5}}}{5} + \frac{{{x^7}}}{{7\; \times \;3}}} \right]_0^1\)
\(= \frac{1}{5} + \frac{1}{{21}} = \frac{{26}}{{105}}\)