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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![The image shows a definite integral from 0 to 1 of the polynomial function \(5x^9 - 15x^4 + 20\) with respect to \(x\).
Mathematically, this is represented as:
\[
\int_{0}^{1} (5x^9 - 15x^4 + 20) \, dx
\]
This integral calculates the area under the curve of the function from \(x = 0\) to \(x = 1\).
To solve this integral, you need to find the antiderivative of the polynomial function and then evaluate it from 0 to 1:
1. **Antiderivative Calculation:**
- The antiderivative of \(5x^9\) is \(\frac{5}{10}x^{10} = \frac{1}{2}x^{10}\).
- The antiderivative of \(-15x^4\) is \(-\frac{15}{5}x^5 = -3x^5\).
- The antiderivative of \(20\) is \(20x\).
2. **Evaluate from 0 to 1:**
After finding the antiderivative, substitute the upper limit of 1 and the lower limit of 0, subtracting the latter from the former:
\[
\left[\frac{1}{2}x^{10} - 3x^5 + 20x\right]_{0}^{1}
\]
This will provide the definite integral, giving the exact area under the curve from 0 to 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70ffcee4-44fb-4130-b7f2-60fe2df8c3c8%2F3f0686b1-0df8-43b1-b1d6-4536572f101e%2F3bzniao.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a definite integral from 0 to 1 of the polynomial function \(5x^9 - 15x^4 + 20\) with respect to \(x\).
Mathematically, this is represented as:
\[
\int_{0}^{1} (5x^9 - 15x^4 + 20) \, dx
\]
This integral calculates the area under the curve of the function from \(x = 0\) to \(x = 1\).
To solve this integral, you need to find the antiderivative of the polynomial function and then evaluate it from 0 to 1:
1. **Antiderivative Calculation:**
- The antiderivative of \(5x^9\) is \(\frac{5}{10}x^{10} = \frac{1}{2}x^{10}\).
- The antiderivative of \(-15x^4\) is \(-\frac{15}{5}x^5 = -3x^5\).
- The antiderivative of \(20\) is \(20x\).
2. **Evaluate from 0 to 1:**
After finding the antiderivative, substitute the upper limit of 1 and the lower limit of 0, subtracting the latter from the former:
\[
\left[\frac{1}{2}x^{10} - 3x^5 + 20x\right]_{0}^{1}
\]
This will provide the definite integral, giving the exact area under the curve from 0 to 1.
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