Question

Calcule a integral definida de função y=x4 no intervalo entre (0) e (5/2) e escolha a alternativa CORRETA. 0 437/333 30/32 625/32 2034/33

Answer 1

Minha resposta deu 625/32

Answer 2

✅ Após resolver os cálculos, concluímos que a integral definida no referido intervalo de integração é:

  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \int_{0}^{\frac{5}{2}} x^{4}\,dx = \frac{625}{32}\:u\cdot a\:\:\:}}\end{gathered} }$

   

Seja a função:        

                 $\Large\displaystyle\begin{gathered}\tt y = x^{4}\end{gathered}$

Podemos reescreve-la como:

             $\Large\displaystyle\begin{gathered}\tt f(x) = x^{4}\end{gathered}$  

Se queremos calcular a integral de "f(x)" no intervalo "I":

           $\Large\displaystyle\begin{gathered}\tt I = \left[0,\:5/2\right]\end{gathered}$

Então, temos:

        $\Large\displaystyle\text{ \begin{gathered}\tt \int_{0}^{\frac{5}{2}} x^{4}\,dx = \bigg(\frac{x^{4 + 1}}{4 + 1} + c\Bigg)\Bigg|_{0}^{\frac{5}{2}}\end{gathered} }$

                            $\Large\displaystyle\text{ \begin{gathered}\tt = \bigg(\frac{x^{5}}{5} + c\Bigg)\Bigg|_{0}^{\frac{5}{2}}\end{gathered} }$

                            $\Large\displaystyle\text{ \begin{gathered}\tt = \Bigg(\frac{(\frac{5}{2})^{5}}{5} + c\Bigg) - \Bigg(\frac{0}{5} + c\Bigg)\end{gathered} }$

                            $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{\frac{3125}{32}}{5} + c - 0 - c\end{gathered} }$

                            $\Large\displaystyle\begin{gathered}\tt = \frac{3125}{32}\cdot\frac{1}{5}\end{gathered}$

                            $\Large\displaystyle\begin{gathered}\tt = \frac{3125\:(\div5)}{160\:(\div5)}\end{gathered}$

                            $\Large\displaystyle\begin{gathered}\tt = \frac{625}{32}\end{gathered}$

Portanto, a integral definida entre os referidos limites é:

         $\Large\displaystyle\text{ \begin{gathered}\tt \int_{0}^{\frac{5}{2}} x^{4}\,dx = \frac{625}{32}\:u\cdot a\end{gathered} }$

Saiba mais:

1. https://brainly.com.br/tarefa/8202825
2. https://brainly.com.br/tarefa/4341918
3. https://brainly.com.br/tarefa/36147694

Solução gráfica: