Solve any maths problem

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1. \int_{0}^{1} 2000 \pi r \left(1 - r^{2}\right)\, dr

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

2. \int 2000 \pi r \left(1 - r^{2}\right)\, dr = 2000 \pi \int r \left(1 - r^{2}\right)\, dr

   Pull the constant 2000 \pi out of the integral.

3. u = r^{2},\quad du = 2 r\, dr

   Substitute u = r^{2}.

4. \int r \left(1 - r^{2}\right)\, dr = \int \frac{1}{2} - \frac{u}{2}\, d_u

   Rewrite the integral in terms of u.

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

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

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

   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{r^{4}}{4} + \frac{r^{2}}{2}

   Substitute back u = r^{2}.

10. F(1) - F(0) = \left(500 \pi\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

11. = 500 \pi \approx 1570.8

    Simplify.