Integrate e^(x) from -3 to 4: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{-3}^{4} {e}^{x}\,dx$
2. 2. Step by step
   1. $$\int_{-3}^{4} {e}^{x}\,dx$$
      Start with the definite integral
   2. $$\int {e}^{x}\,dx = {e}^{x}$$
      Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
   3. $$F(x) = {e}^{x}$$
      Antiderivative
   4. $$F(4) - F(-3) = 54.59815003 - 0.04978706837 = 54.54836296$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{-3}^{4} {e}^{x}\,dx = 54.54836296$$

   ∫ from -3 to 4 = 54.54836296

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 54.54836296 ✓

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