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Definite integralsCalculusCollegeabout 1:16

Integrate 3x^2 - 2x from -1 to 2: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 6.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫−123x2−2x dx\int_{-1}^{2} 3{x}^{2}-2x\,dx
  2. 2. Steps, part 1
    1. ∫−123x2−2x dx\int_{-1}^{2} 3{x}^{2}-2x\,dx
      Start with the definite integral
    2. ∫x2 dx=x33\int {x}^{2}\,dx = \frac{{x}^{3}}{3}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    3. ∫3x2 dx=x3\int 3{x}^{2}\,dx = {x}^{3}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    4. ∫x dx=x22\int x\,dx = \frac{{x}^{2}}{2}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
  3. 3. Steps, part 2
    1. ∫2x dx=x2\int 2x\,dx = {x}^{2}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    2. ∫3x2−2x dx=x3−x2\int 3{x}^{2}-2x\,dx = {x}^{3}-{x}^{2}
      Sum rule: integrate term by term
    3. F(x)=x3−x2F(x) = {x}^{3}-{x}^{2}
      Antiderivative
    4. F(2)−F(−1)=4−(−2)=6F(2) - F(-1) = 4 - \left(-2\right) = 6
      Apply the fundamental theorem of calculus
  4. 4. The answer
    ∫−123x2−2x dx=6\int_{-1}^{2} 3{x}^{2}-2x\,dx = 6

    ∫ from -1 to 2 = 6

  5. 5. Check
    • Differentiated the antiderivative and compared it with the integrand ✓
    • Adaptive Simpson's rule gives 6 ✓

Transcript

The problem. Here is the problem: integrate 3x squared minus 2x from negative 1 to 2. Evaluate the definite integral of 3 x squared minus 2 x from x equals negative 1 to x equals 2.

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). Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

Steps, part 2. Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Sum rule: integrate term by term. Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 6. 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 6, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

More lessons like this

Integrate 3x^2 - 2x from -1 to 2: Definite integrals Lesson