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Definite integralsCalculusCollegeabout 1:09

Integrate x^3-2x from 0 to 4: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 48.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫04x3−2x dx\int_{0}^{4} {x}^{3}-2x\,dx
  2. 2. Steps, part 1
    1. ∫04x3−2x dx\int_{0}^{4} {x}^{3}-2x\,dx
      Start with the definite integral
    2. ∫x3 dx=x44\int {x}^{3}\,dx = \frac{{x}^{4}}{4}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    3. ∫x dx=x22\int x\,dx = \frac{{x}^{2}}{2}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    4. ∫2x dx=x2\int 2x\,dx = {x}^{2}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
  3. 3. Steps, part 2
    1. ∫x3−2x dx=x44−x2\int {x}^{3}-2x\,dx = \frac{{x}^{4}}{4}-{x}^{2}
      Sum rule: integrate term by term
    2. F(x)=x44−x2F(x) = \frac{{x}^{4}}{4}-{x}^{2}
      Antiderivative
    3. F(4)−F(0)=48−0=48F(4) - F(0) = 48 - 0 = 48
      Apply the fundamental theorem of calculus
  4. 4. The answer
    ∫04x3−2x dx=48\int_{0}^{4} {x}^{3}-2x\,dx = 48

    ∫ from 0 to 4 = 48

  5. 5. Check
    • Differentiated the antiderivative and compared it with the integrand ✓
    • Adaptive Simpson's rule gives 48 ✓

Transcript

The problem. Here is the problem: integrate x cubed minus 2x from 0 to 4. Evaluate the definite integral of x cubed minus 2 x from x equals 0 to x equals 4.

Steps, part 1. Start with the definite integral. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1). Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1). Constant multiple rule: the integral of k times f dx equals k the integral of f dx.

Steps, part 2. Sum rule: integrate term by term. Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 48. 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.

Check. Differentiated the antiderivative and compared it with the integrand, which checks out. Adaptive Simpson prime s rule gives 48, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

More lessons like this

Integrate x^3-2x from 0 to 4: Animated Step-by-Step Lesson