Definite integral of sin(x)^2 from 0 to 2pi: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{2\pi} {\sin\left( x\right)}^{2}\,dx$
2. 2. Step by step
   1. $$\int_{0}^{2\pi} {\sin\left( x\right)}^{2}\,dx$$
      Start with the definite integral
   2. $$\int {\sin\left( x\right)}^{2}\,dx = \frac{x}{2}-\frac{\sin\left(2x\right)}{4}$$
      Power-reducing identity: sin² u = (1 − cos 2u)/2
   3. $$F(x) = \frac{x}{2}-\frac{\sin\left(2x\right)}{4}$$
      Antiderivative
   4. $$F(2\pi) - F(0) = 3.141592654 - 0 = 3.141592654$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{2\pi} {\sin\left( x\right)}^{2}\,dx = 3.141592654$$

   ∫ from 0 to 2π = 3.141592654 = π

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 3.141592654 ✓

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