Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Problem 1: Evaluate the Integral Consider the double integral: \[ \int_{0}^{1} \int_{-\sqrt{1-x^2}}^{0} 2x \cos \left( y - \frac{y^3}{3} \right) \, dy \, dx \] #### Explanation: This problem involves evaluating a double integral over a specified region. The inner integral is with respect to \( y \), ranging from \( -\sqrt{1-x^2} \) to \( 0 \). The outer integral is with respect to \( x \), ranging from \( 0 \) to \( 1 \). The integrand consists of the function \( 2x \cos \left( y - \frac{y^3}{3} \right) \), incorporating both the trigonometric cosine function and a cubic polynomial in \( y \). The problem likely requires utilizing integration techniques suitable for such functions, potentially involving substitution or numerical methods, depending on the context. #### Steps for Evaluation: 1. **Start with the inner integral**: Integrate \( 2x \cos \left( y - \frac{y^3}{3} \right) \) with respect to \( y \). 2. **Proceed to the outer integral**: Once the inner integration is performed, integrate the resulting expression with respect to \( x \). This problem illustrates a typical application of double integrals, emphasizing the importance of understanding the limits of integration and the integration techniques applicable to the given functions.
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Transcribed Image Text:### Problem 1: Evaluate the Integral Consider the double integral: \[ \int_{0}^{1} \int_{-\sqrt{1-x^2}}^{0} 2x \cos \left( y - \frac{y^3}{3} \right) \, dy \, dx \] #### Explanation: This problem involves evaluating a double integral over a specified region. The inner integral is with respect to \( y \), ranging from \( -\sqrt{1-x^2} \) to \( 0 \). The outer integral is with respect to \( x \), ranging from \( 0 \) to \( 1 \). The integrand consists of the function \( 2x \cos \left( y - \frac{y^3}{3} \right) \), incorporating both the trigonometric cosine function and a cubic polynomial in \( y \). The problem likely requires utilizing integration techniques suitable for such functions, potentially involving substitution or numerical methods, depending on the context. #### Steps for Evaluation: 1. **Start with the inner integral**: Integrate \( 2x \cos \left( y - \frac{y^3}{3} \right) \) with respect to \( y \). 2. **Proceed to the outer integral**: Once the inner integration is performed, integrate the resulting expression with respect to \( x \). This problem illustrates a typical application of double integrals, emphasizing the importance of understanding the limits of integration and the integration techniques applicable to the given functions.
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