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8. Identify the definite integral that represents the area of the surface formed by revolving the graph of f(x) = x² on the interval [0, 2] about the x-axis. *277 5.0²2 +2² (a) 2π (d) 2π + [²³» √ x²√√1 + x² dx y 1 + = dy y (a) 2π 277 S x√² + 9x³² dx Jo √√√2 [² (b) 2π (e) None of these x²√1 + 4x² dx (b) 2π 9dentify the definite integral that represents the area of the surface formed by revolving the graph of f(x) = x³ on the interval [0, 1] about the y-axis. f. - 27 for TV √1 + 9x dx (c) 2π x√1 + 4x² dx (c) 2π * x² √/1 + 3x²³dx Jo

8. Identify the definite integral that represents the area of the surface formed by revolving the graph of f(x) = x² on the interval [0, 2] about the x-axis. *277 5.0²2 +2² (a) 2π (d) 2π + [²³» √ x²√√1 + x² dx y 1 + = dy y (a) 2π 277 S x√² + 9x³² dx Jo √√√2 [² (b) 2π (e) None of these x²√1 + 4x² dx (b) 2π 9dentify the definite integral that represents the area of the surface formed by revolving the graph of f(x) = x³ on the interval [0, 1] about the y-axis. f. - 27 for TV √1 + 9x dx (c) 2π x√1 + 4x² dx (c) 2π * x² √/1 + 3x²³dx Jo

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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**Chapter: Applications of Integration** **7.** Write the definite integral that represents the arc length of one period of the curve \( y = \sin 2x \). (Do not evaluate the integral) **Answer:** \[ \int_0^{\pi} \sqrt{1 + 4 \cos^2 2x} \, dx \] --- **8.** Identify the definite integral that represents the area of the surface formed by revolving the graph of \( f(x) = x^2 \) on the interval \([0, \sqrt{2}]\) about the x-axis. - (a) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + x^2} \, dx\) - (b) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x^2 \, dx\) ⟶ **Circled answer** - (c) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + 4x^2} \, dx\) - (d) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + 9x^2} \, dx\) - (e) None of these **9.** Identify the definite integral that represents the area of the surface formed by revolving the graph of \( f(x) = x^3 \) on the interval \([0, 1]\) about the y-axis. - (a) \(\displaystyle 2 \pi \int_0^1 x \sqrt{1 + 9x^2} \, dx\) - (b) \(\displaystyle 2 \pi \int_0^1 x^2 \, dx\) ⟶ **Circled answer** - (c) \(\displaystyle 2 \pi \int_0^1 x^2 \sqrt{1 + 4x^2} \, dx\) - (d) None of these **Notes on the Problems:** - These problems require identifying the correct integral forms for arc lengths and surfaces of revolution, critical skills in applied calculus. - Pay attention to the functions and the axes about which the revolutions occur, as they determine the setup of the integrals.
Transcribed Image Text:**Chapter: Applications of Integration** **7.** Write the definite integral that represents the arc length of one period of the curve \( y = \sin 2x \). (Do not evaluate the integral) **Answer:** \[ \int_0^{\pi} \sqrt{1 + 4 \cos^2 2x} \, dx \] --- **8.** Identify the definite integral that represents the area of the surface formed by revolving the graph of \( f(x) = x^2 \) on the interval \([0, \sqrt{2}]\) about the x-axis. - (a) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + x^2} \, dx\) - (b) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x^2 \, dx\) ⟶ **Circled answer** - (c) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + 4x^2} \, dx\) - (d) \(\displaystyle 2 \pi \int_0^{\sqrt{2}} x \sqrt{1 + 9x^2} \, dx\) - (e) None of these **9.** Identify the definite integral that represents the area of the surface formed by revolving the graph of \( f(x) = x^3 \) on the interval \([0, 1]\) about the y-axis. - (a) \(\displaystyle 2 \pi \int_0^1 x \sqrt{1 + 9x^2} \, dx\) - (b) \(\displaystyle 2 \pi \int_0^1 x^2 \, dx\) ⟶ **Circled answer** - (c) \(\displaystyle 2 \pi \int_0^1 x^2 \sqrt{1 + 4x^2} \, dx\) - (d) None of these **Notes on the Problems:** - These problems require identifying the correct integral forms for arc lengths and surfaces of revolution, critical skills in applied calculus. - Pay attention to the functions and the axes about which the revolutions occur, as they determine the setup of the integrals.
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