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11. Evaluate the following integrals: (a) 2+ 2- dt 1/2 (b) ["a 1 dt 2. (c) I12-x| dx

11. Evaluate the following integrals: (a) 2+ 2- dt 1/2 (b) ["a 1 dt 2. (c) I12-x| dx

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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**Educational Content: Integrals and Particle Motion** ### Integral Problems and Particle Motion Analysis #### 11. Evaluate the following integrals: - (a) \(\int \sqrt{2 + \sqrt{2 - t}} \, dt\) - (b) \(\int_0^1 \frac{t}{\sqrt{1 - t^2}} \, dt\) - (c) \(\int_2^3 12 - x \, dx\) - (d) \(\int_{\ln 2}^{\ln 3} e^{2x} \, dx\) **Notes and Solutions:** - For the given integrals, attempts are shown with steps and simplifications. - There are handwritten solutions and calculations provided next to each integral problem. #### 12. Particle Motion Problem A particle travels along a line. Its velocity in meters per second is given by \( v(t) = 3t^2 - 12t \). Find: - (a) The displacement from \( t = 0 \) to \( t = 8 \). - (b) The distance traveled by the particle from \( t = 0 \) to \( t = 8 \). **Graphical Explanation:** - Graphs or diagrams are not provided, but understanding the function of velocity and its integration is crucial for determining displacement (using definite integrals) and distance traveled (using the absolute value of the integrals). **Handwritten Calculations:** - Next to the problems are solutions that involve step-by-step integration and simplification. Transcendental functions and properties of logarithmic or exponential functions might be employed. Note the attempts to evaluate definite integrals with boundaries as shown in the solutions. This educational module covers fundamental integral evaluations and applications in particle motion, essential for physics and mathematics students.
Transcribed Image Text:**Educational Content: Integrals and Particle Motion** ### Integral Problems and Particle Motion Analysis #### 11. Evaluate the following integrals: - (a) \(\int \sqrt{2 + \sqrt{2 - t}} \, dt\) - (b) \(\int_0^1 \frac{t}{\sqrt{1 - t^2}} \, dt\) - (c) \(\int_2^3 12 - x \, dx\) - (d) \(\int_{\ln 2}^{\ln 3} e^{2x} \, dx\) **Notes and Solutions:** - For the given integrals, attempts are shown with steps and simplifications. - There are handwritten solutions and calculations provided next to each integral problem. #### 12. Particle Motion Problem A particle travels along a line. Its velocity in meters per second is given by \( v(t) = 3t^2 - 12t \). Find: - (a) The displacement from \( t = 0 \) to \( t = 8 \). - (b) The distance traveled by the particle from \( t = 0 \) to \( t = 8 \). **Graphical Explanation:** - Graphs or diagrams are not provided, but understanding the function of velocity and its integration is crucial for determining displacement (using definite integrals) and distance traveled (using the absolute value of the integrals). **Handwritten Calculations:** - Next to the problems are solutions that involve step-by-step integration and simplification. Transcendental functions and properties of logarithmic or exponential functions might be employed. Note the attempts to evaluate definite integrals with boundaries as shown in the solutions. This educational module covers fundamental integral evaluations and applications in particle motion, essential for physics and mathematics students.
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