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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Concept explainers
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.
Question
Find the area of the enclosed by

Transcribed Image Text:**Problem Statement:**
**Find the area enclosed by** $y = 1 + \sqrt{x}$ **and** $y = \frac{4 + x}{4}$.
The given mathematical task involves finding the area between two curves. Here are the equations of the curves:
1. \( y = 1 + \sqrt{x} \)
2. \( y = \frac{4 + x}{4} \)
To solve this problem, one would usually proceed with the following steps:
1. **Find the Points of Intersection:**
Determine where the two curves intersect by setting the equations equal to each other and solving for \( x \).
2. **Set Up the Integral:**
Identify the region that lies between the curves. The integral will be set up with the difference between the functions over the range bounded by their points of intersection.
3. **Evaluate the Integral:**
Compute the definite integral to find the area between the two curves.
If a graph or diagram is present, it would show the plots of both functions intersecting at specific points, illustrating the enclosed area that needs to be calculated.
**Note for Educators:** Graphical visualizations help students understand the relationship between the functions and the concept of the area between curves.
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