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Definite integralsCalculusCollegeabout 0:44

Integrate abs(x) from -1 to 3: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 5.

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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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int_{-1}^{3} \left| x\right|\,dx$
2. Step by step
1. $\int_{-1}^{3} \left| x\right|\,dx$
  Start with the definite integral
2. $\int \left| x\right|\,dx = \frac{x\cdot\left| x\right|}{2}$
  ∫ |u| du = u|u|/2
3. $F(x) = \frac{x\cdot\left| x\right|}{2}$
  Antiderivative
4. $F(3) - F(-1) = 4.5 - \left(-0.5\right) = 5$
  Apply the fundamental theorem of calculus
3. The answer
$\int_{-1}^{3} \left| x\right|\,dx = 5$
∫ from -1 to 3 = 5
4. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives 5 ✓

Transcript

The problem. Here is the problem: integrate abs(x) from negative 1 to 3. Evaluate the definite integral of abs(x) from x equals negative 1 to x equals 3.

Step by step. Start with the definite integral. the integral of |u| du equals u|u|/2. Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 5. 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 5, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Integrate abs(x) from -1 to 3: Animated Step-by-Step Lesson