Multiple Choice
Find the exact length of the curve for .
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8. Definite Integrals
Introduction to Definite Integrals
3:57 minutesProblem 8.1.24
Textbook Question
7–64. Integration review Evaluate the following integrals.
24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx
Verified step by step guidance1
Step 1: Simplify the integrand. The given integrand is \( \frac{x^{5/2} - x^{1/2}}{x^{3/2}} \). Use the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \) to simplify each term: \( \frac{x^{5/2}}{x^{3/2}} = x^{(5/2 - 3/2)} = x^1 \) and \( \frac{x^{1/2}}{x^{3/2}} = x^{(1/2 - 3/2)} = x^{-1} \). The integrand simplifies to \( x - x^{-1} \).
Step 2: Set up the integral with the simplified integrand. The integral becomes \( \int_{0}^{\theta} (x - x^{-1}) \, dx \).
Step 3: Split the integral into two separate terms. Using the linearity of integration, rewrite the integral as \( \int_{0}^{\theta} x \, dx - \int_{0}^{\theta} x^{-1} \, dx \).
Step 4: Compute the antiderivative of each term. For \( \int x \, dx \), the antiderivative is \( \frac{x^2}{2} \). For \( \int x^{-1} \, dx \), the antiderivative is \( \ln|x| \). Substitute these results into the integral: \( \left[ \frac{x^2}{2} \right]_{0}^{\theta} - \left[ \ln|x| \right]_{0}^{\theta} \).
Step 5: Apply the limits of integration. Substitute \( x = \theta \) and \( x = 0 \) into the antiderivatives. Be cautious with \( \ln|x| \) at \( x = 0 \), as it is undefined. This suggests the integral may need further consideration for convergence at the lower limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the accumulation of a function's values over a specific interval, represented as ∫ from a to b f(x) dx. It provides the net area under the curve of the function f(x) between the limits a and b. In this question, the integral is evaluated from 0 to θ, indicating the area under the curve from x = 0 to x = θ.
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Simplifying Rational Functions
Rational functions are ratios of polynomials, and simplifying them often involves factoring and reducing terms. In the given integral, the expression (x⁵/2 - x¹/2) / x³/2 can be simplified by dividing each term in the numerator by the denominator. This simplification is crucial for making the integral easier to evaluate.
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Power Rule for Integration
The power rule for integration states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. This rule is essential for integrating polynomial functions, as it allows for straightforward computation of the integral. In this problem, applying the power rule will help in finding the antiderivative of the simplified function before evaluating the definite integral.
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