Question

Hi there,I need help with this Fundamental Theorem of Calculus question.

Answer 1

To find the integral we need to divide it according to the intervals that define the piece-wise function, then we have:

`\int_0^2f(x)dx=\int_0^1f(x)dx+\int_1^2f(x)dx`

Now that we have divide the integral we plug the expressions that define it in the intervals:

`\begin{gathered} \int_0^2f(x)dx=\int_0^16x^4dx+\int_1^23x^5dx \\ =((6)/(5)x^5)\vert_0^1+((3)/(6)x^6)\vert_1^2 \\ =((6)/(5)(1)^5-(6)/(5)(0)^5)+((3)/(6)(2)^6-(3)/(6)(1)^6) \\ =(6)/(5)+32-(1)/(2) \\ =(327)/(10) \end{gathered}`

Therefore, the integral is 327/10