Thể tích đất sét cần sử dụng là:

\(V = \pi \int\limits_0^{31} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} - \pi \int\limits_0^{30} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} \)

\[ = \pi \int\limits_0^{30} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} + \pi \int\limits_{30}^{31} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} - \pi \int\limits_0^{30} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} \]

\[ = \pi \int\limits_{30}^{31} {{{\left( {\frac{1}{x^2} + \frac{3}x + 5} \right)}^2}dx} \]

\[ = \pi \int\limits_{30}^{31} {\left( {\frac{1}{{{{175}^2}}}{x^4} + \frac{9}{x^2} + 25 + \frac{6}{x^3} + \frac{2}{x^2} + \frac{6}{7}x} \right)dx} \]

\[ = \pi \int\limits_{30}^{31} {\left( {\frac{1}{{{{175}^2}}}{x^4} + \frac{6}{x^3} + \frac{x^2} + \frac{6}{7}x + 25} \right)dx} \]

\[ = \pi \left. {\left( {\frac{{{x^5}}} + \frac{{3{x^4}}} + \frac{{79{x^3}}} + \frac{{3{x^2}}}{7} + 25x} \right)} \right|_{30}^{31}\]

\[ \approx \pi \left( {2240,4 - 2073,2} \right) = 167,2\pi \] (cm³).

Vậy để hoàn thành bình gốm đó ta cần sử dụng 167,2π cm³ đất sét.