10.5 3.5 10.5 Let + [10 f(x) dx = 9, [." *F(x) dx = 2, and [... (a) [F(x f(x) dx = 4 7 (6f(x) — 5) dx = | - (b) S (6f(x) X f(x) dx = 3. Find the following.
10.5 3.5 10.5 Let + [10 f(x) dx = 9, [." *F(x) dx = 2, and [... (a) [F(x f(x) dx = 4 7 (6f(x) — 5) dx = | - (b) S (6f(x) X f(x) dx = 3. Find the following.
Calculus: Early Transcendentals
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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![### Understanding Definite Integrals
Let:
\[ \int_{0}^{10.5} f(x) \, dx = 9, \]
\[ \int_{0}^{3.5} f(x) \, dx = 2, \]
and
\[ \int_{7}^{10.5} f(x) \, dx = 3. \]
Find the following:
#### (a) Evaluate \( \int_{3.5}^{7} f(x) \, dx \):
The given information informs us that:
\[ \int_{0}^{10.5} f(x) \, dx = \int_{0}^{3.5} f(x) \, dx + \int_{3.5}^{7} f(x) \, dx + \int_{7}^{10.5} f(x) \, dx. \]
Given:
\[ \int_{0}^{10.5} f(x) \, dx = 9, \]
\[ \int_{0}^{3.5} f(x) \, dx = 2, \]
and
\[ \int_{7}^{10.5} f(x) \, dx = 3. \]
We can find \( \int_{3.5}^{7} f(x) \, dx \) by rearranging the equation:
\[ 9 = 2 + \int_{3.5}^{7} f(x) \, dx + 3. \]
Solving for \( \int_{3.5}^{7} f(x) \, dx \):
\[ \int_{3.5}^{7} f(x) \, dx = 9 - 2 - 3, \]
\[ \int_{3.5}^{7} f(x) \, dx = 4. \]
Thus,
\[ \int_{3.5}^{7} f(x) \, dx = 4. \]
This value is given correctly, as indicated by the green check mark.
#### (b) Evaluate \( \int_{3.5}^{7} (6f(x) - 5) \, dx \):
We can use the linearity of the integral to separate the terms:
\[ \int_{3.5}^{7} (6f(x) - 5) \, dx = 6 \int_{3.5}^{7} f(x) \, dx - \int_{3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F926c3fce-ea0b-442b-8045-2d4039975256%2Fc44f77f8-95fd-4282-91c5-e5137c919d7c%2Fqvpswq9_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Definite Integrals
Let:
\[ \int_{0}^{10.5} f(x) \, dx = 9, \]
\[ \int_{0}^{3.5} f(x) \, dx = 2, \]
and
\[ \int_{7}^{10.5} f(x) \, dx = 3. \]
Find the following:
#### (a) Evaluate \( \int_{3.5}^{7} f(x) \, dx \):
The given information informs us that:
\[ \int_{0}^{10.5} f(x) \, dx = \int_{0}^{3.5} f(x) \, dx + \int_{3.5}^{7} f(x) \, dx + \int_{7}^{10.5} f(x) \, dx. \]
Given:
\[ \int_{0}^{10.5} f(x) \, dx = 9, \]
\[ \int_{0}^{3.5} f(x) \, dx = 2, \]
and
\[ \int_{7}^{10.5} f(x) \, dx = 3. \]
We can find \( \int_{3.5}^{7} f(x) \, dx \) by rearranging the equation:
\[ 9 = 2 + \int_{3.5}^{7} f(x) \, dx + 3. \]
Solving for \( \int_{3.5}^{7} f(x) \, dx \):
\[ \int_{3.5}^{7} f(x) \, dx = 9 - 2 - 3, \]
\[ \int_{3.5}^{7} f(x) \, dx = 4. \]
Thus,
\[ \int_{3.5}^{7} f(x) \, dx = 4. \]
This value is given correctly, as indicated by the green check mark.
#### (b) Evaluate \( \int_{3.5}^{7} (6f(x) - 5) \, dx \):
We can use the linearity of the integral to separate the terms:
\[ \int_{3.5}^{7} (6f(x) - 5) \, dx = 6 \int_{3.5}^{7} f(x) \, dx - \int_{3
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