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2. Evaluate the improper integrals or show that they diverge. Make careful use of limit notation. (a) 10 2 2 √x-2 (b) (7 2x dz + 2)² ze-2 dr dz

2. Evaluate the improper integrals or show that they diverge. Make careful use of limit notation. (a) 10 2 2 √x-2 (b) (7 2x dz + 2)² ze-2 dr dz

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Calculus Problems #### Limits 1. Evaluate the following limits: (a) \(\lim_{x \to \infty} \frac{1}{x}\) (b) \(\lim_{x \to \infty} \frac{x^5}{e^x}\) #### Improper Integrals 2. Evaluate the improper integrals or show that they diverge. Make careful use of limit notation. (a) \(\int_{2}^{10} \frac{2}{\sqrt{x-2}} \, dx\) (b) \(\int_{-\infty}^{0} \frac{2x}{(x^2 + 2)^2} \, dx\) (c) \(\int_{0}^{\infty} x e^{-2x} \, dx\) #### Convergence Testing with the Comparison Test 3. Use the Comparison Test to determine whether or not the integral converges. (a) \(\int_{3}^{\infty} \frac{dx}{\sqrt{x-1}}\) (b) \(\int_{0}^{5} \frac{dx}{x^{1/3} + x^3}\) (c) \(\int_{1}^{\infty} \frac{1}{\sqrt{x^2 + 2}} \, dx\) #### Additional Calculus Problems 4. Find the arc length of \(y = 2x\sqrt{x}\), from \(x = 0\) to \(x = 1\). 5. Find the centroid of the region enclosed by the parabola \(y = 1 - x^2\) and by the x-axis.
Transcribed Image Text:### Calculus Problems #### Limits 1. Evaluate the following limits: (a) \(\lim_{x \to \infty} \frac{1}{x}\) (b) \(\lim_{x \to \infty} \frac{x^5}{e^x}\) #### Improper Integrals 2. Evaluate the improper integrals or show that they diverge. Make careful use of limit notation. (a) \(\int_{2}^{10} \frac{2}{\sqrt{x-2}} \, dx\) (b) \(\int_{-\infty}^{0} \frac{2x}{(x^2 + 2)^2} \, dx\) (c) \(\int_{0}^{\infty} x e^{-2x} \, dx\) #### Convergence Testing with the Comparison Test 3. Use the Comparison Test to determine whether or not the integral converges. (a) \(\int_{3}^{\infty} \frac{dx}{\sqrt{x-1}}\) (b) \(\int_{0}^{5} \frac{dx}{x^{1/3} + x^3}\) (c) \(\int_{1}^{\infty} \frac{1}{\sqrt{x^2 + 2}} \, dx\) #### Additional Calculus Problems 4. Find the arc length of \(y = 2x\sqrt{x}\), from \(x = 0\) to \(x = 1\). 5. Find the centroid of the region enclosed by the parabola \(y = 1 - x^2\) and by the x-axis.
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