Evaluate $\int_0^1 x^2\,dx$.

How we read your problem

Evaluate the definite integral of x² from x = 0 to x = 1.

Step-by-step solution

1. 1

   Start with the definite integral

   $$\int_{0}^{1} {x}^{2}\,dx$$
2. 2

   Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)

   $$\int {x}^{2}\,dx = \frac{{x}^{3}}{3}$$
3. 3

   Antiderivative

   $$F(x) = \frac{{x}^{3}}{3}$$
4. 4

   Apply the fundamental theorem of calculus

   $$F(1) - F(0) = 0.3333333333 - 0 = \frac{1}{3} \approx 0.33333333$$

Formulas used

• Power rule

  $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\ n \neq -1$$

• Fundamental theorem of calculus

  $$\int_a^b f(x)\,dx = F(b) - F(a)$$

A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

Verification

• Differentiated the antiderivative and compared it with the integrand ✓
• Adaptive Simpson's rule gives 0.3333333333 ✓