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Definite integralsCalculusCollegeabout 0:47

Integrate x^2 from 0 to 3: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 9.

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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
$\int_{0}^{3} {x}^{2}\,dx$
2. Step by step
1. $\int_{0}^{3} {x}^{2}\,dx$
  Start with the definite integral
2. $\int {x}^{2}\,dx = \frac{{x}^{3}}{3}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. $F(x) = \frac{{x}^{3}}{3}$
  Antiderivative
4. $F(3) - F(0) = 9 - 0 = 9$
  Apply the fundamental theorem of calculus
3. The answer
$\int_{0}^{3} {x}^{2}\,dx = 9$
∫ from 0 to 3 = 9
4. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives 9 ✓

Transcript

The problem. Here is the problem: integrate x squared from 0 to 3. Evaluate the definite integral of x squared from x equals 0 to x equals 3.

Step by step. 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). Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 9. 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 9, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

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