le 1,0 A graph of f(x) is shown above. Using the geometry of the graph, evaluate the definite integrals. a) f(x) dx = b) f(x) dx = c) f(x) dx = d) f(x) dx = 10 e) 3f(x) dx =
le 1,0 A graph of f(x) is shown above. Using the geometry of the graph, evaluate the definite integrals. a) f(x) dx = b) f(x) dx = c) f(x) dx = d) f(x) dx = 10 e) 3f(x) dx =
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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Riemann Integral
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![### Evaluating Definite Integrals Using Geometry
#### Problem Statement
A graph of \( f(x) \) is shown below. Using the geometry of the graph, evaluate the following definite integrals:
a) \[ \int_4^{10} f(x) \, dx = \]
b) \[ \int_0^4 f(x) \, dx = \]
c) \[ \int_0^{10} f(x) \, dx = \]
d) \[ \int_3^4 f(x) \, dx = \]
e) \[ \int_3^{10} f(x) \, dx = \]
#### Graph Explanation
The given graph appears to show a linear function declining from a certain point on the \( y \)-axis and passing through another point, forming a right triangle beneath the \( x \)-axis. Key points and specific values on the graph are:
- Starts at \( y = 4 \) when \( x = 0 \).
- Crosses the \( x \)-axis at \( x = 8 \).
- Ends at \( y = -2 \) when \( x = 10 \).
##### Steps to Evaluate
1. **Calculate the area of geometric shapes (triangle and rectangle) influenced by the curve.**
2. **Use appropriate limits of integration to determine the areas under the curve.**
Using the geometric interpretation for a triangle:
\[ \text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
a) **From \( x = 4 \) to \( x = 10 \):**
- Analyze the part of the triangle from \( x = 4 \) to \( x = 8 \), using height 4, and part from \( x = 8 \) to \( x = 10 \) with a height extending to -2.
b) **From \( x = 0 \) to \( x = 4 \):**
- This area's height starts at 4 and decreases linearly.
c) **From \( x = 0 \) to \( x = 10 \):**
- Includes entire triangular region below initially positive \( y \) values crossing negative \( y \) values.
d) **From \( x = 3 \) to \( x = 4 \):**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa090e79e-b63b-42aa-a337-95657eeca289%2F6e0c036b-4819-4507-b59e-3277f0893a27%2Fgl71wkh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating Definite Integrals Using Geometry
#### Problem Statement
A graph of \( f(x) \) is shown below. Using the geometry of the graph, evaluate the following definite integrals:
a) \[ \int_4^{10} f(x) \, dx = \]
b) \[ \int_0^4 f(x) \, dx = \]
c) \[ \int_0^{10} f(x) \, dx = \]
d) \[ \int_3^4 f(x) \, dx = \]
e) \[ \int_3^{10} f(x) \, dx = \]
#### Graph Explanation
The given graph appears to show a linear function declining from a certain point on the \( y \)-axis and passing through another point, forming a right triangle beneath the \( x \)-axis. Key points and specific values on the graph are:
- Starts at \( y = 4 \) when \( x = 0 \).
- Crosses the \( x \)-axis at \( x = 8 \).
- Ends at \( y = -2 \) when \( x = 10 \).
##### Steps to Evaluate
1. **Calculate the area of geometric shapes (triangle and rectangle) influenced by the curve.**
2. **Use appropriate limits of integration to determine the areas under the curve.**
Using the geometric interpretation for a triangle:
\[ \text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
a) **From \( x = 4 \) to \( x = 10 \):**
- Analyze the part of the triangle from \( x = 4 \) to \( x = 8 \), using height 4, and part from \( x = 8 \) to \( x = 10 \) with a height extending to -2.
b) **From \( x = 0 \) to \( x = 4 \):**
- This area's height starts at 4 and decreases linearly.
c) **From \( x = 0 \) to \( x = 10 \):**
- Includes entire triangular region below initially positive \( y \) values crossing negative \( y \) values.
d) **From \( x = 3 \) to \( x = 4 \):**
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