Integrate |x - 1| from 0 to 3: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{3} \left| x-1\right|\,dx$
2. 2. Step by step
   1. $$\int_{0}^{3} \left| x-1\right|\,dx$$
      Start with the definite integral
   2. $$\int \left| x-1\right|\,dx = \frac{\left( x-1\right)\cdot\left| x-1\right|}{2}$$
      ∫ |u| du = u|u|/2
   3. $$F(x) = \frac{\left( x-1\right)\cdot\left| x-1\right|}{2}$$
      Antiderivative
   4. $$F(3) - F(0) = 2 - \left(-0.5\right) = \frac{5}{2} \approx 2.5$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{3} \left| x-1\right|\,dx = \frac{5}{2} \approx 2.5$$

   ∫ from 0 to 3 = 5/2 ≈ 2.5

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 2.5 ✓

Evaluate definite integrals with the fundamental theorem of calculus.

• Video lesson

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

  ▶0:51
  CalculusCollege4 steps

  Integrate x/(x²+1) from 0 to 2

  Definite integrals

• Video lesson

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

  ▶0:51
  CalculusCollege4 steps

  Integrate x/(x²+1) from 0 to 1

  Definite integrals

• Video lesson

  $\int_{0}^{3} {x}^{2}\,dx$

  ▶0:46
  CalculusCollege4 steps

  Integrate x² from 0 to 3

  Definite integrals

• Video lesson

  $\int_{-3}^{4} {x}^{2} \sin\left( x\right)\,dx$

  ▶0:59
  CalculusCollege5 steps

  Integrate x²sin(x) from -3 to 4

  Definite integrals

• Video lesson

  $\int_{0}^{\pi} \sin\left( x\right)\,dx$

  ▶0:43
  CalculusCollege4 steps

  Integral of sin(x) from 0 to π

  Definite integrals

• Video lesson

  $\int_{-3}^{0} \cos\left( x\right)\,dx$

  ▶0:44
  CalculusCollege4 steps

  Integrate cos(x) from -3 to 0

  Definite integrals

• All definite integrals lessons
• Browse the whole lesson library
• Make a lesson from your own problem
• Step-by-step solver: check your own answer.