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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Question
![Evaluate the integral:
\[
\int_{1}^{\sqrt{2}} \frac{9}{1 + x^2} \, dx
\]
(Note: The integral expression is written in a rectangular box, indicating an area for writing or calculating the result.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b638ef2-9928-44ee-907b-a6bc7b9add08%2F68190756-42bc-4b78-9f8e-5ef69cc97fe7%2Fe228cmh_processed.png&w=3840&q=75)
Transcribed Image Text:Evaluate the integral:
\[
\int_{1}^{\sqrt{2}} \frac{9}{1 + x^2} \, dx
\]
(Note: The integral expression is written in a rectangular box, indicating an area for writing or calculating the result.)
![**Evaluate the Definite Integral**
\[
\int_{4}^{7} (15x^2 - 8x + 7) \, dx
\]
---
In this problem, you are asked to evaluate the definite integral of the polynomial function \(15x^2 - 8x + 7\) from the lower limit \(x = 4\) to the upper limit \(x = 7\). Definite integrals represent the signed area under a curve described by a function within the given limits.
To solve this integral, follow these steps:
1. **Find the Indefinite Integral:**
\[
\int (15x^2 - 8x + 7) \, dx
\]
- Use the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\).
- Integrate each term separately:
- \(\int 15x^2 \, dx = 15 \cdot \frac{x^{3}}{3} = 5x^3\)
- \(\int -8x \, dx = -8 \cdot \frac{x^{2}}{2} = -4x^2\)
- \(\int 7 \, dx = 7x\)
- Combine the results:
\[ F(x) = 5x^3 - 4x^2 + 7x + C \]
2. **Evaluate the Definite Integral:**
\[
\int_{4}^{7} (15x^2 - 8x + 7) \, dx = F(7) - F(4)
\]
- Calculate \(F(7)\):
\[ F(7) = 5(7)^3 - 4(7)^2 + 7(7) \]
- Calculate \(F(4)\):
\[ F(4) = 5(4)^3 - 4(4)^2 + 7(4) \]
- Subtract the results:
\[ F(7) - F(4) \]
By evaluating this expression, you will obtain the numerical result representing the signed area under the curve \(15x^2 - 8x + 7\) from \(x = 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b638ef2-9928-44ee-907b-a6bc7b9add08%2F68190756-42bc-4b78-9f8e-5ef69cc97fe7%2Frcd10gy_processed.png&w=3840&q=75)
Transcribed Image Text:**Evaluate the Definite Integral**
\[
\int_{4}^{7} (15x^2 - 8x + 7) \, dx
\]
---
In this problem, you are asked to evaluate the definite integral of the polynomial function \(15x^2 - 8x + 7\) from the lower limit \(x = 4\) to the upper limit \(x = 7\). Definite integrals represent the signed area under a curve described by a function within the given limits.
To solve this integral, follow these steps:
1. **Find the Indefinite Integral:**
\[
\int (15x^2 - 8x + 7) \, dx
\]
- Use the power rule for integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\).
- Integrate each term separately:
- \(\int 15x^2 \, dx = 15 \cdot \frac{x^{3}}{3} = 5x^3\)
- \(\int -8x \, dx = -8 \cdot \frac{x^{2}}{2} = -4x^2\)
- \(\int 7 \, dx = 7x\)
- Combine the results:
\[ F(x) = 5x^3 - 4x^2 + 7x + C \]
2. **Evaluate the Definite Integral:**
\[
\int_{4}^{7} (15x^2 - 8x + 7) \, dx = F(7) - F(4)
\]
- Calculate \(F(7)\):
\[ F(7) = 5(7)^3 - 4(7)^2 + 7(7) \]
- Calculate \(F(4)\):
\[ F(4) = 5(4)^3 - 4(4)^2 + 7(4) \]
- Subtract the results:
\[ F(7) - F(4) \]
By evaluating this expression, you will obtain the numerical result representing the signed area under the curve \(15x^2 - 8x + 7\) from \(x = 4
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