Risolvi qualsiasi problema di matematica

Passo dopo passo

1. \int_{0}^{2} 2 \pi x \left(4 - x^{2}\right)\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. \int 2 \pi x \left(4 - x^{2}\right)\, dx = 2 \pi \int x \left(4 - x^{2}\right)\, dx

   Pull the constant 2 \pi out of the integral.

3. u = x^{2},\quad du = 2 x\, dx

   Substitute u = x^{2}.

4. \int x \left(4 - x^{2}\right)\, dx = \int 2 - \frac{u}{2}\, d_u

   Rewrite the integral in terms of u.

5. \int 2 - \frac{u}{2}\, d_u = \int 2\, d_u + \int - \frac{u}{2}\, d_u

   The integral of a sum is the sum of the integrals.

6. \int 2\, d_u = 2 u

   The integral of a constant c is c·x.

7. \int - \frac{u}{2}\, d_u = - \frac{1}{2} \int u\, d_u

   Pull the constant - \frac{1}{2} out of the integral.

8. \int u\, d_u = \frac{u^{2}}{2}

   Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

9. = - \frac{x^{4}}{4} + 2 x^{2}

   Substitute back u = x^{2}.

10. F(2) - F(0) = \left(8 \pi\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

11. = 8 \pi \approx 25.133

    Simplify.