Domanda del libro di testo
Finding surface area
Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.AAE.5
Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))
Guida verificata passo dopo passo1
Start by examining the integral: \(\int \frac{dt}{t - \sqrt{1 - t^{2}}}\). Notice the expression in the denominator involves both \(t\) and \(\sqrt{1 - t^{2}}\), which suggests a trigonometric substitution might simplify the square root term.
Use the substitution \(t = \sin(\theta)\), which implies \(dt = \cos(\theta) d\theta\). This substitution is helpful because \(\sqrt{1 - t^{2}}\) becomes \(\sqrt{1 - \sin^{2}(\theta)} = \cos(\theta)\).
Rewrite the integral in terms of \(\theta\): replace \(t\) with \(\sin(\theta)\) and \(dt\) with \(\cos(\theta) d\theta\). The integral becomes \(\int \frac{\cos(\theta) d\theta}{\sin(\theta) - \cos(\theta)}\).
To simplify the integral, consider dividing numerator and denominator by \(\cos(\theta)\) (assuming \(\cos(\theta) \neq 0\)), which transforms the integral into \(\int \frac{d\theta}{\tan(\theta) - 1}\). This form is easier to handle.
Next, use the substitution \(u = \tan(\theta) - 1\), so that \(du = \sec^{2}(\theta) d\theta\). Express \(d\theta\) in terms of \(du\) and rewrite the integral accordingly. This will allow you to integrate with respect to \(u\) and then back-substitute to \(t\).
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Integration Techniques
Integration techniques involve methods such as substitution, partial fractions, or trigonometric substitution to simplify and evaluate integrals. Recognizing the form of the integrand helps determine the appropriate technique to apply for solving the integral efficiently.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Trigonometric Substitution
Trigonometric substitution is used to simplify integrals involving expressions like √(1 - t²) by substituting t = sin(θ) or t = cos(θ). This transforms the integral into a trigonometric form that is often easier to integrate.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Algebraic Manipulation of the Integrand
Algebraic manipulation involves rewriting the integrand to a simpler or more recognizable form, such as rationalizing the denominator or combining terms. This step is crucial to make the integral more approachable for standard integration methods.
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05:22Completing the Square to Rewrite the Integrand
Pratica correlata
Domanda del libro di testo
Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.
lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)
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18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.
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Centroid of a region
Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.
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Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.
∫ csc θ dθ
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Domanda del libro di testo
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)
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