Integrate abs(x) from -2 to -1: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{-2}^{-1} \left| x\right|\,dx$
2. 2. Step by step
   1. $$\int_{-2}^{-1} \left| x\right|\,dx$$
      Start with the definite integral
   2. $$\int \left| x\right|\,dx = \frac{x\cdot\left| x\right|}{2}$$
      ∫ |u| du = u|u|/2
   3. $$F(x) = \frac{x\cdot\left| x\right|}{2}$$
      Antiderivative
   4. $$F(-1) - F(-2) = -0.5 - \left(-2\right) = \frac{3}{2} \approx 1.5$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{-2}^{-1} \left| x\right|\,dx = \frac{3}{2} \approx 1.5$$

   ∫ from -2 to -1 = 3/2 ≈ 1.5

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 1.5 ✓

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