'5 Given f(x) dx = 16 and f(x) dx = 8, evaluate (a) f(x) dx. 0. (b) f(x) dx. 5 (c) f(x) dx. '5 (d) 2f(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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Given:

\[
\int_0^5 f(x) \, dx = 16 \quad \text{and} \quad \int_5^7 f(x) \, dx = 8
\]

Evaluate:

(a) \(\int_0^7 f(x) \, dx\).

(b) \(\int_5^0 f(x) \, dx\).

(c) \(\int_5^5 f(x) \, dx\).

(d) \(\int_0^5 2f(x) \, dx\).

### Explanation:

1. **Problem (a):** Combine the integrals from 0 to 5 and from 5 to 7 to find the integral from 0 to 7.

2. **Problem (b):** Reverse the limits of integration for the integral from 5 to 0, which will change the sign.

3. **Problem (c):** The integral with identical upper and lower limits is always zero.

4. **Problem (d):** Use the property of integrals involving constants to evaluate the integral with the function multiplied by 2.
Transcribed Image Text:Given: \[ \int_0^5 f(x) \, dx = 16 \quad \text{and} \quad \int_5^7 f(x) \, dx = 8 \] Evaluate: (a) \(\int_0^7 f(x) \, dx\). (b) \(\int_5^0 f(x) \, dx\). (c) \(\int_5^5 f(x) \, dx\). (d) \(\int_0^5 2f(x) \, dx\). ### Explanation: 1. **Problem (a):** Combine the integrals from 0 to 5 and from 5 to 7 to find the integral from 0 to 7. 2. **Problem (b):** Reverse the limits of integration for the integral from 5 to 0, which will change the sign. 3. **Problem (c):** The integral with identical upper and lower limits is always zero. 4. **Problem (d):** Use the property of integrals involving constants to evaluate the integral with the function multiplied by 2.
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