maths.free › Calculus › Applications of integration: area, volume, arc length

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.

Contoh soal · 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

Langkah demi langkah

  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.

Tampilkan jawaban
8 \pi

Sekarang Anda Pilih soal, atau ketik atau gambar soal Anda sendiri. Setiap langkah, sebuah gambar, jawabannya tersembunyi sampai Anda memintanya.

Simpan pekerjaan Anda sendiri

Sebuah akun gratis menambahkan catatan pada setiap pelajaran, catatan apa yang telah Anda selesaikan, masalah yang Anda selesaikan di satu tempat, dan tutor yang dapat Anda tanyakan tentang halaman ini. matematika itu sendiri terbuka untuk semua orang, ditandatangani atau tidak.

Daftar Masuk

Simbol yang digunakan di sini

Sentuh simbol apapun untuk definisi lengkap, gambar, dan apa setiap huruf di dalamnya berarti.

Bagaimana: 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.

Pertanyaan yang sering diajukan

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.

Lainnya di Calculus

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