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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![On this webpage, we will walk through the evaluation of a definite integral. The integral we will work on is:
\[ \int_{3}^{5} (15x^2 - 4x + 3)\,dx \]
To solve this definite integral, follow these steps:
1. **Integrate the function** \(15x^2 - 4x + 3\):
\[
\int (15x^2 - 4x + 3)\,dx = \int 15x^2\,dx - \int 4x\,dx + \int 3\,dx
\]
\[
= 15 \int x^2\,dx - 4 \int x\,dx + 3 \int\,dx
\]
2. **Apply the power rule** for integration:
\[
15 \int x^2\,dx = 15 \left( \frac{x^3}{3} \right) = 5x^3
\]
\[
-4 \int x\,dx = -4 \left( \frac{x^2}{2} \right) = -2x^2
\]
\[
3 \int\,dx = 3x
\]
3. **Combine the integrated terms**:
\[
\int (15x^2 - 4x + 3)\,dx = 5x^3 - 2x^2 + 3x + C
\]
4. **Evaluate the integral from the limits** 3 to 5:
\[
\left[ 5x^3 - 2x^2 + 3x \right]_{3}^{5}
\]
5. **Substitute the limits** into the integrated function:
\[
\left(5(5)^3 - 2(5)^2 + 3(5)\right) - \left(5(3)^3 - 2(3)^2 + 3(3)\right)
\]
Simplifying each part:
\[
= \left( 5 \cdot 125 - 2 \cdot 25 + 3 \cdot 5 \right) - \left( 5 \cdot 27 - 2 \cdot 9 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dc9f679-6754-47d7-a5b7-cef477361a1e%2Fe3eb1bb7-2317-4c8b-a414-d524dfab7d77%2F6vvgpe5_processed.png&w=3840&q=75)
Transcribed Image Text:On this webpage, we will walk through the evaluation of a definite integral. The integral we will work on is:
\[ \int_{3}^{5} (15x^2 - 4x + 3)\,dx \]
To solve this definite integral, follow these steps:
1. **Integrate the function** \(15x^2 - 4x + 3\):
\[
\int (15x^2 - 4x + 3)\,dx = \int 15x^2\,dx - \int 4x\,dx + \int 3\,dx
\]
\[
= 15 \int x^2\,dx - 4 \int x\,dx + 3 \int\,dx
\]
2. **Apply the power rule** for integration:
\[
15 \int x^2\,dx = 15 \left( \frac{x^3}{3} \right) = 5x^3
\]
\[
-4 \int x\,dx = -4 \left( \frac{x^2}{2} \right) = -2x^2
\]
\[
3 \int\,dx = 3x
\]
3. **Combine the integrated terms**:
\[
\int (15x^2 - 4x + 3)\,dx = 5x^3 - 2x^2 + 3x + C
\]
4. **Evaluate the integral from the limits** 3 to 5:
\[
\left[ 5x^3 - 2x^2 + 3x \right]_{3}^{5}
\]
5. **Substitute the limits** into the integrated function:
\[
\left(5(5)^3 - 2(5)^2 + 3(5)\right) - \left(5(3)^3 - 2(3)^2 + 3(3)\right)
\]
Simplifying each part:
\[
= \left( 5 \cdot 125 - 2 \cdot 25 + 3 \cdot 5 \right) - \left( 5 \cdot 27 - 2 \cdot 9 +
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