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
![If
\[
\int_{-94}^{-11} f(x) \, dx = 8
\]
and
\[
\int_{-94}^{-11} g(x) \, dx = 10
\]
and
\[
\int_{-94}^{-11} h(x) \, dx = 23,
\]
what does the following integral equal?
\[
\int_{-94}^{-11} [5f(x) + 2g(x) - h(x)] \, dx = \, \boxed{}
\]
**Explanation:**
To solve the integral
\[
\int_{-94}^{-11} [5f(x) + 2g(x) - h(x)] \, dx,
\]
we can use the linearity property of integrals. This allows us to separate the integral into individual components:
\[
= 5 \int_{-94}^{-11} f(x) \, dx + 2 \int_{-94}^{-11} g(x) \, dx - \int_{-94}^{-11} h(x) \, dx
\]
Substitute the given values:
\[
= 5(8) + 2(10) - 23
\]
\[
= 40 + 20 - 23
\]
\[
= 37
\]
So, the value of the integral is \(\boxed{37}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9cd79cd-6f4d-4477-9046-9d8af5db1a08%2F880e53ff-eff0-4588-853b-6f010c0e74e7%2Fbeols8j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If
\[
\int_{-94}^{-11} f(x) \, dx = 8
\]
and
\[
\int_{-94}^{-11} g(x) \, dx = 10
\]
and
\[
\int_{-94}^{-11} h(x) \, dx = 23,
\]
what does the following integral equal?
\[
\int_{-94}^{-11} [5f(x) + 2g(x) - h(x)] \, dx = \, \boxed{}
\]
**Explanation:**
To solve the integral
\[
\int_{-94}^{-11} [5f(x) + 2g(x) - h(x)] \, dx,
\]
we can use the linearity property of integrals. This allows us to separate the integral into individual components:
\[
= 5 \int_{-94}^{-11} f(x) \, dx + 2 \int_{-94}^{-11} g(x) \, dx - \int_{-94}^{-11} h(x) \, dx
\]
Substitute the given values:
\[
= 5(8) + 2(10) - 23
\]
\[
= 40 + 20 - 23
\]
\[
= 37
\]
So, the value of the integral is \(\boxed{37}\).
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