Calculus intergrals

Question

Calculate 𝐹(8) given that 𝐹(2)=8 and 𝐹′(𝑥)=𝑥^2. Hint: Express 𝐹(8)−𝐹(2) as a definite integral.

(Use symbolic notation and fractions where needed.)

Answer 1

According to the Fundamental Theorem of Calculus, ∫_{(from a to b)}F'(x)dx = F(b) - F(a), where a < b.

F(8) = F(2) + F(8) - F(2)

By the Fundamental Theorem of Calculus, we have F(8) = F(2) + ∫_{(from 2 to 8)} F'(x)dx

So, F(8) = 8 + ∫_{(from 2 to 8)} x²dx = 8 + (1/3)x³_{(from 2 to 8)}

Evaluate and simplify.