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Chapter 5: Problem 14

Find the area of the region bounded by the given graphs. \(y=x^{2}+1, y=x^{2}, x=1, x=3\)

Short Answer

Expert verified
The area is 2.

Step by step solution

01

Identify the Graphs

Identify and plot the given functions: \( y = x^2 + 1 \), \( y = x^2 \), and the vertical lines \( x = 1 \) and \( x = 3 \.\)
02

Set Up the Integral

The area between the two curves \( y = x^2 + 1 \) and \( y = x^2 \) bounded by \( x = 1 \) and \( x = 3 \) is found by integrating the difference of the functions over the interval \( [1, 3] \.\)
03

Form the Integral

Set up the definite integral: \[ \text{Area} = \int_{1}^{3} [(x^2 + 1) - x^2] \, dx \]
04

Simplify the Integrand

Simplify the integrand inside the integral: \[ \text{Area} = \int_{1}^{3} (x^2 + 1 - x^2) \, dx = \int_{1}^{3} 1 \, dx \]
05

Evaluate the Integral

Evaluate the integral: \[ \text{Area} = \, x \bigg|_{1}^{3} = 3 - 1 = 2 \]

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Key Concepts

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

Definite Integral
To find the area between curves, we utilize the concept of a definite integral. A definite integral helps calculate the area under a curve within specified bounds.
The notation for a definite integral looks like this: \[ \text{Area} = \int_{a}^{b} f(x) \, dx \]
Here, \(a\) and \(b\) are the bounds on the \(x\)-axis.
We apply this concept by setting up an integral with a function representing the height of the area at any point \(x\). For the given problem, we integrate between \(1\) and \(3\), with our integrand being \(1\), representing the height between the curves. The evaluation of this definite integral gives us the exact area enclosed.
Integration
Integration is a fundamental concept in calculus, and it's essential for finding areas, volumes, and other quantities that accumulate.In the problem, integration is used to compute an area between two curves. The steps involve:
  • Setting up the integral.
  • Simplifying the integrand.
  • Evaluating the integral.
For instance, we set up the problem as:
\[\begin{equation*} \int_{1}^{3} [(x^2 + 1) - x^2] \, dx \end{equation*}\]
After simplifying inside the integrand, we integrate the function \( 1 \). Once done, we find the definite integral鈥檚 value, which, in this case, yields an area of \(2\). This process shows why understanding integration deeply is crucial for problems involving areas between curves.
Bounded Region
When dealing with areas between curves, the bounded region is crucial.A bounded region is the area enclosed by the given curves and specified boundaries.In the exercise, the bounded region is between:
  • \( y = x^2 + 1 \)
  • \( y = x^2 \)
  • Vertical lines \(x = 1\) and \(x = 3\)
To sketch this region, you would plot the curves and lines to visualize where they intersect and the area trapped between them.
Understanding how to identify and visualize the bounded region is the first essential step before setting up the integral. This makes it easier to comprehend the limits within which you will integrate and ensures accurate calculation of the enclosed area.

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