Find the area between the graph of y = x - 6x + 8 and the x-axis from x = 1 to x = 3.

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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**Problem: Calculating Area Under a Curve**

Find the area between the graph of \( y = x^3 - 6x + 8 \) and the x-axis from \( x = 1 \) to \( x = 3 \).

**Solution Approach**

To determine the area between the curve and the x-axis over the specified interval, you need to calculate the definite integral of the function \( y = x^3 - 6x + 8 \) from \( x = 1 \) to \( x = 3 \).

1. **Set Up the Integral:**

   \[
   \int_{1}^{3} (x^3 - 6x + 8) \, dx
   \]

2. **Compute the Integral:**

   - The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \).
   - The antiderivative of \( -6x \) is \( -3x^2 \).
   - The antiderivative of \( 8 \) is \( 8x \).

3. **Evaluate the Definite Integral:**

   \[
   \left[ \frac{x^4}{4} - 3x^2 + 8x \right]_{1}^{3}
   \]

   - Substitute \( x = 3 \) and \( x = 1 \) into the antiderivative and subtract to find the accumulated area between the curve and the x-axis over the given interval.

This procedure provides the required area under the curve from \( x = 1 \) to \( x = 3 \).
Transcribed Image Text:**Problem: Calculating Area Under a Curve** Find the area between the graph of \( y = x^3 - 6x + 8 \) and the x-axis from \( x = 1 \) to \( x = 3 \). **Solution Approach** To determine the area between the curve and the x-axis over the specified interval, you need to calculate the definite integral of the function \( y = x^3 - 6x + 8 \) from \( x = 1 \) to \( x = 3 \). 1. **Set Up the Integral:** \[ \int_{1}^{3} (x^3 - 6x + 8) \, dx \] 2. **Compute the Integral:** - The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \). - The antiderivative of \( -6x \) is \( -3x^2 \). - The antiderivative of \( 8 \) is \( 8x \). 3. **Evaluate the Definite Integral:** \[ \left[ \frac{x^4}{4} - 3x^2 + 8x \right]_{1}^{3} \] - Substitute \( x = 3 \) and \( x = 1 \) into the antiderivative and subtract to find the accumulated area between the curve and the x-axis over the given interval. This procedure provides the required area under the curve from \( x = 1 \) to \( x = 3 \).
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