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Chapter 5: Problem 2

Find the area under the given curve over the indicated interval. $$ y=5 ; \quad[1,3] $$

Short Answer

Expert verified
The area under the curve y=5 from x=1 to x=3 is 10.

Step by step solution

01

Understand the Problem

The task is to find the area under the curve given by the function \( y=5 \) over the interval \([1,3]\).
02

Identify the Function and Interval

The function is \( y = 5 \), which is a horizontal line. The interval is from \( x = 1 \) to \( x = 3 \).
03

Calculate the Length of the Interval

The interval length is found by subtracting the lower limit from the upper limit: \( \text{Length of the Interval} = 3 - 1 = 2 \)
04

Use the Area Formula for a Rectangle

Since the function is a constant, the area under the curve forms a rectangle. Use the formula for the area of a rectangle: \( \text{Area} = \text{height} \times \text{width} \) Here, the height is the value of the function \( y = 5 \) and the width is the length of the interval \( 2 \).
05

Calculate the Area

Multiply the height by the width to get the area: \( \text{Area} = 5 \times 2 = 10 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular area
To find the area under a curve, we sometimes simplify our problem by recognizing shapes we are already familiar with, such as rectangles. In this particular exercise, we are given a horizontal line represented by the equation \( y = 5 \), meaning that for any value of \( x \) in the interval \([1,3]\), the value of \( y \) stays constant at 5.

The area under this line can be visualized as a rectangle because the function is constant - it doesn't change. The shape formed is a simple rectangle where:
  • The height is the constant value of the function, which is 5.
  • The width is the length of the interval along the x-axis, from 1 to 3.
By using the formula for the area of a rectangle (\( \text{Area} = \text{height} \times \text{width} \)), we can easily calculate the area under the curve.
Interval length
The length of the interval is crucial for finding the area under a constant function. Imagine you are walking along a path from point A to point B. The distance you cover is the interval length.

In our exercise, the interval is given as \([1,3]\). To find its length, we subtract the starting point (lower limit) from the ending point (upper limit):
\

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