Question

Determine a área aproximada entre a função g(x) = 2x² - 18 e o eixo x, sabendo que o valor da abscissa varia de 4 a 5.

Answer 1

✅ Após resolver os cálculos, concluímos que a área procurada é:

  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf S = \int_{4}^{5}(2x^{2} - 18)\,dx = \frac{68}{3}\,u.\,a.\:\:\:}}\end{gathered} }$

Sejam os dados:

            $\Large\begin{cases}\tt g(x) = 2x^{2} - 18\\ \tt a = 4\\ \tt b = 5\end{cases}$

Para calcular a área "S" entre a função polinomial e o intervalo referido devemos calcular a integral definida no referido intervalo.

Sabendo que o TFC - Teorema Fundamental do Cálculo, diz:

 $\Large\displaystyle\begin{gathered}\tt S = \int_{a}^{b}g(x)\,dx = G(x)|_{a}^{b} = G(b) - G(a)\end{gathered}$

Então, temos:

     $\Large\displaystyle\begin{gathered}\tt S = \int_{a}^{b}g(x)\,dx\end{gathered}$

         $\Large\displaystyle\begin{gathered}\tt = \int_{4}^{5}(2x^{2} - 18)\,dx\end{gathered}$

         $\Large\displaystyle\text{ \begin{gathered}\tt = \bigg(\frac{2x^{2 + 1}}{2 + 1} - 18x + c\bigg)\bigg|_{4}^{5}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered}\tt = \bigg(\frac{2x^{3}}{3} - 18x + c\bigg)\bigg|_{4}^{5}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered}\tt = \bigg(\frac{2\cdot5^{3}}{3} - 18\cdot5 + c\bigg) - \bigg(\frac{2\cdot4^{3}}{3} - 18\cdot4 + c\bigg)\end{gathered} }$

         $\Large\displaystyle\begin{gathered}\tt = \bigg(\frac{250}{3} - 90 + c\bigg) - \bigg(\frac{128}{3} - 72 + c\bigg)\end{gathered}$

         $\Large\displaystyle\begin{gathered}\tt = \frac{250}{3} - 90 + c - \frac{128}{3} + 72 - c\end{gathered}$

         $\Large\displaystyle\begin{gathered}\tt = \frac{250}{3} - 90 - \frac{128}{3} + 72\end{gathered}$

         $\Large\displaystyle\begin{gathered}\tt = \frac{250 - 270 - 128 + 216}{3}\end{gathered}$

         $\Large\displaystyle\begin{gathered}\tt = \frac{68}{3}\end{gathered}$

Portanto, a área é:

    $\Large\displaystyle\begin{gathered}\tt S = \frac{68}{3}\,u.\,a.\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$

Answer 2

Resposta:

9,89

Explicação passo a passo: