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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.
مثال کار شده · integrate pi*(sqrt(x))^2 dx from 0 to 4
قدم به قدم
- \int_{0}^{4} \pi x\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
- \int \pi x\, dx = \pi \int x\, dx
Pull the constant \pi out of the integral.
- \int x\, dx = \frac{x^{2}}{2}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
- F(4) - F(0) = \left(8 \pi\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
- = 8 \pi \approx 25.133
Simplify.
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چطور: Applications of integration: area, volume, arc length
- First find an antiderivative F, then evaluate F(b) − F(a).
- Pull the constant \pi out of the integral.
- Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
- Fundamental theorem of calculus: plug in the limits.
- Simplify.
سوالاتي که مردم ميپرسن
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.
بیشتر در Calculus
LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsInfinite series and convergence tests