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Calculus · Definite-Integrals

Integrate pi·r·(1/2 - r^2/2) from 0 to 1

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

Step by step

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

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

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

    Pull the constant \pi out of the integral.

  3. u = \frac{1}{2} - \frac{r^{2}}{2},\quad du = - r\, dr

    Substitute u = \frac{1}{2} - \frac{r^{2}}{2}.

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

    Rewrite the integral in terms of u.

  5. \int - u\, d_u = -1 \int u\, d_u

    Pull the constant -1 out of the integral.

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

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

  7. = - \frac{\left(\frac{1}{2} - \frac{r^{2}}{2}\right)^{2}}{2}

    Substitute back u = \frac{1}{2} - \frac{r^{2}}{2}.

  8. F(1) - F(0) = \left(0\right) - \left(- \frac{\pi}{8}\right)

    Fundamental theorem of calculus: plug in the limits.

  9. = \frac{\pi}{8} \approx 0.39270

    Simplify.

Reveal the answer
\frac{\pi}{8}