Definite integral of sqrt(x) from 1 to 9: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{9} \sqrt{x}\,dx$
2. 2. Step by step
   1. $$\int_{1}^{9} \sqrt{x}\,dx$$
      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. $$F(9) - F(1) = 18 - 0.6666666667 = \frac{52}{3} \approx 17.333333$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{1}^{9} \sqrt{x}\,dx = \frac{52}{3} \approx 17.333333$$

   ∫ from 1 to 9 = 52/3 ≈ 17.333333

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 17.33333333 ✓

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