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Calculus · Definite-Integrals

Integrate 1.2·e^(-z/8000) from 0 to 8000

\int_{0}^{8000} 1.2 e^{- \frac{z}{8000}}\, dz

Step by step

  1. \int_{0}^{8000} 1.2 e^{- \frac{z}{8000}}\, dz

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

  2. \int 1.2 e^{- \frac{z}{8000}}\, dz = 1.2 \int e^{- \frac{z}{8000}}\, dz

    Pull the constant 1.2 out of the integral.

  3. u = - \frac{z}{8000},\quad du = - \frac{1}{8000}\, dz

    Substitute u = - \frac{z}{8000}.

  4. \int e^{- \frac{z}{8000}}\, dz = \int - 8000 e^{u}\, d_u

    Rewrite the integral in terms of u.

  5. \int - 8000 e^{u}\, d_u = -8000 \int e^{u}\, d_u

    Pull the constant -8000 out of the integral.

  6. \int e^{u}\, d_u = e^{u}

    ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

  7. = - 8000 e^{- \frac{z}{8000}}

    Substitute back u = - \frac{z}{8000}.

  8. F(8000) - F(0) = \left(- \frac{9600.0}{e}\right) - \left(-9600.0\right)

    Fundamental theorem of calculus: plug in the limits.

  9. = 9600.0 - \frac{9600.0}{e} \approx 6068.4

    Simplify.

Reveal the answer
9600.0 - \frac{9600.0}{e}