Question

Can anyone help me solve this? (image) (Calculus 2 - Indefinite Integrals)

Answer 1

recall \(\displaystyle f(b)-f(a)=\int_a^b f'(x)\ dx\)

Answer 2

so $$f(b)=f(a)+\int_a^bf'(x)\ dx\\f(x)=f(0)+\int_0^x f'(x)\ dx$$

Answer 3

now interpret $$\int_0^x f'(x)\ dx$$to be the (signed) area bounded by the function given

Answer 4

sorry, I'm confused here?

Answer 5

We are trying to find the anti derivative correct?

Answer 6

do you know what the fundamental theorem of calculus us? we're not finding an antiderivative we're computing the function \(f\) at different points knowing the graph of its derivative

Answer 7

I remember learning it, but can't recall atm. I don't really understand it.

Answer 8

e.g. since \(f(0)=3\) we know:$$f(1)=3+\int_0^1 f'(x)\ dx$$we can interpret this integral geometrically as the area of a triangular piece: