Integral of sqrt(x) from 0 to 4: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{4} \sqrt{x}\,dx$
2. 2. Step by step
   1. $$\int_{0}^{4} \sqrt{x}\,dx$$
      Start with the definite integral
   2. $$\int \sqrt{x}\,dx = \frac{{x}^{\frac{3}{2}}\cdot2}{3}$$
      ∫ √x dx = (2/3)x^(3/2)
   3. $$F(x) = \frac{{x}^{\frac{3}{2}}\cdot2}{3}$$
      Antiderivative
   4. $$F(4) - F(0) = 5.333333333 - 0 = \frac{16}{3} \approx 5.3333333$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{4} \sqrt{x}\,dx = \frac{16}{3} \approx 5.3333333$$

   ∫ from 0 to 4 = 16/3 ≈ 5.3333333

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 5.333333333 ✓

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