Answered: 6 = √² (6x - x²) dx = We now have Area…

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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$**Calculating the Area under the Curve** To find the area under the curve defined by the function $6x - x^2$ from $x = 0$ to $x = 6$, we use the definite integral: \[ \text{We now have Area} = \int_{0}^{6} (6x - x^2) \, dx \] The integral sign signifies that we are summing up infinitesimal areas under the function $6x - x^2$ over the interval from 0 to 6. ### Steps to Solve: 1. **Expand the Integrand:** \[ 6x - x^2 \] 2. **Integrate the Function:** The integral of $6x - x^2$ will be computed, and it involves basic integration rules. 3. **Evaluate the Integral:** \[ \left[ \int (6x - x^2) \, dx \right]_{0}^{6} \] This notation indicates that we will first find the antiderivative of the integrand, and then evaluate it at the upper and lower limits, subtracting the values accordingly. 4. **Final Area Calculation:** Insert the antiderivative’s value evaluated at $x = 6$ and $x = 0$ and subtract them. \[ = \left[ \text{Antiderivative} \right]_0^6 \] Here, the antiderivative part is left to be computed and substituted in the limit values to get the area. **Explanation of Symbols:** - $\int$: Integral symbol indicates summation. - $0$ and $6$ under and above the integral sign are the lower and upper limits of integration, respectively. - $dx$: Indicates integration with respect to $x$. - The red cross at the bottom indicates a step yet to be completed in the process. **Note:** The detailed steps of integrating the function and finding the antiderivative are crucial for understanding how to complete this calculation accurately.$

Transcribed Image Text:**Calculating the Area under the Curve** To find the area under the curve defined by the function $6x - x^2$ from $x = 0$ to $x = 6$, we use the definite integral: \[ \text{We now have Area} = \int_{0}^{6} (6x - x^2) \, dx \] The integral sign signifies that we are summing up infinitesimal areas under the function $6x - x^2$ over the interval from 0 to 6. ### Steps to Solve: 1. **Expand the Integrand:** \[ 6x - x^2 \] 2. **Integrate the Function:** The integral of $6x - x^2$ will be computed, and it involves basic integration rules. 3. **Evaluate the Integral:** \[ \left[ \int (6x - x^2) \, dx \right]_{0}^{6} \] This notation indicates that we will first find the antiderivative of the integrand, and then evaluate it at the upper and lower limits, subtracting the values accordingly. 4. **Final Area Calculation:** Insert the antiderivative’s value evaluated at $x = 6$ and $x = 0$ and subtract them. \[ = \left[ \text{Antiderivative} \right]_0^6 \] Here, the antiderivative part is left to be computed and substituted in the limit values to get the area. **Explanation of Symbols:** - $\int$: Integral symbol indicates summation. - $0$ and $6$ under and above the integral sign are the lower and upper limits of integration, respectively. - $dx$: Indicates integration with respect to $x$. - The red cross at the bottom indicates a step yet to be completed in the process. **Note:** The detailed steps of integrating the function and finding the antiderivative are crucial for understanding how to complete this calculation accurately.

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