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 Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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![### Integral Evaluation Exercise
#### Given Information:
Suppose the following integrals are known:
\[ \int_{2}^{4} f(x) \, dx = -3 \]
\[ \int_{2}^{7} f(x) \, dx = -6 \]
\[ \int_{2}^{7} g(x) \, dx = 3 \]
#### Task:
Evaluate the following integral:
\[ \int_{7}^{2} g(x) \, dx \]
#### Solution:
(Simplify your answer by considering the properties of definite integrals)
**Note:** Remember that reversing the limits of integration changes the sign of the integral.
\[
\int_{7}^{2} g(x) \, dx = -\int_{2}^{7} g(x) \, dx
\]
Given that:
\[
\int_{2}^{7} g(x) \, dx = 3
\]
Therefore:
\[
\int_{7}^{2} g(x) \, dx = -3
\]
Please substitute the correct answer in the provided box.
[Answer Box: \(_ _ _\)]
**Educational Tip:** In definite integrals, changing the order of the limits (upper to lower and lower to upper) multiplies the integral result by -1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a1263ae-327d-4428-a0aa-137e6771516c%2Ff31dad38-ecab-4498-a43c-09eae9f986b3%2Fn2l491b_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral Evaluation Exercise
#### Given Information:
Suppose the following integrals are known:
\[ \int_{2}^{4} f(x) \, dx = -3 \]
\[ \int_{2}^{7} f(x) \, dx = -6 \]
\[ \int_{2}^{7} g(x) \, dx = 3 \]
#### Task:
Evaluate the following integral:
\[ \int_{7}^{2} g(x) \, dx \]
#### Solution:
(Simplify your answer by considering the properties of definite integrals)
**Note:** Remember that reversing the limits of integration changes the sign of the integral.
\[
\int_{7}^{2} g(x) \, dx = -\int_{2}^{7} g(x) \, dx
\]
Given that:
\[
\int_{2}^{7} g(x) \, dx = 3
\]
Therefore:
\[
\int_{7}^{2} g(x) \, dx = -3
\]
Please substitute the correct answer in the provided box.
[Answer Box: \(_ _ _\)]
**Educational Tip:** In definite integrals, changing the order of the limits (upper to lower and lower to upper) multiplies the integral result by -1.
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