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Applications of integration: area, volume, arc length

Areas between curves, solids of revolution, average value.

Every application is the same move: slice the quantity into thin pieces whose size you can write, then add them with an integral. Area between curves is ∫(top − bottom); a solid of revolution is ∫π r² dx (discs); arc length is ∫√(1 + f′²). Picture it: the shaded region under the curve is the integral being computed. Think it: this slicing is the Riemann sum, and measure theory is what makes it rigorous.

Намунаи коркардашуда · integrate pi*(sqrt(x))^2 dx from 0 to 4

Calculus · Definite-Integrals

Integrate pi·x from 0 to 4

\int_{0}^{4} \pi x\, dx

Қадами ба қадам

  1. \int_{0}^{4} \pi x\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \pi x\, dx = \pi \int x\, dx

    Pull the constant \pi out of the integral.

  3. \int x\, dx = \frac{x^{2}}{2}

    Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

  4. F(4) - F(0) = \left(8 \pi\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  5. = 8 \pi \approx 25.133

    Simplify.

Ҷавоби ҷавобро нишон диҳед
8 \pi

Ҳоло шумо Масъаларо интихоб кунед, ё ки онро навишта ё кашед. Ҳар қадам, тасвир, ҷавоб то он даме, ки шумо пурсед, пинҳон мешавад.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Чӣ тавр: Applications of integration: area, volume, arc length

  1. First find an antiderivative F, then evaluate F(b) − F(a).
  2. Pull the constant \pi out of the integral.
  3. Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
  4. Fundamental theorem of calculus: plug in the limits.
  5. Simplify.

Саволҳои маъмул

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

& Иловаи забон Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsInfinite series and convergence tests