Problem 28 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

The area of the region is 2 square units.

Step 1: Understand the Region

The region is bounded by the curves: - Top: \(y = x^2 + 1\)- Bottom: \(y = x^2\)and by the vertical lines \(x = 1\) and \(x = 3\). The area we want is between these two curves, from \(x = 1\) to \(x = 3\).

Step 2: Set Up the Integral

The area between the curves over an interval \([a, b]\) is given by the integral \(\int_a^b [(\text{top curve}) - (\text{bottom curve})] \, dx\). Here, the integral becomes:\[\int_1^3 [(x^2 + 1) - (x^2)] \, dx = \int_1^3 [1] \, dx.\]

Step 3: Simplify the Integral

Simplifying the expression inside the integral, we get:\[\int_1^3 1 \, dx.\]This simplifies to the integral of a constant, which is a straightforward calculation.

Step 4: Evaluate the Integral

Evaluate the integral:\[\int_1^3 1 \, dx = [x]_1^3 = x \bigg|_1^3 = 3 - 1 = 2.\]

Step 5: Conclusion

The total area of the region bounded by the given graphs is \(2\) square units.

Key Concepts

Definite integral

A definite integral is a key concept in calculus used to find the area under a curve. It can be visualized as the accumulated area from one point (the lower bound) to another (the upper bound) on the x-axis. Imagine drawing slices that stack together closely underneath the curve over a specified interval. This helps you see the area between the curve and the x-axis.

A definite integral \[\int_a^b f(x) \, dx\]is evaluated by calculating the difference between the values of its antiderivative or fundamental function at points \(a\) and \(b\).

The area from \(a\) to \(b\) is calculated with:

• Identify the function \(f(x)\) that describes the curve.
• Determine the limits of integration, \(a\) and \(b\).
• Compute the difference of the antiderivative evaluated at \(b\) and \(a\).

For example, if \(f(x) = 1\), as in this problem between \(x = 1\) and \(x = 3\), the integral results in simply evaluating the difference as \( b - a \), giving you the total area of \(2\) square units.

Bounded region

A bounded region in the context of calculus refers to the enclosed area between curves and/or lines on a graph. This region is often analyzed to determine the size of the area it contains. When talking about a region bounded by multiple functions, we mean finding the space that each curve or line encloses together.

To better visualize this:

• Identify each boundary—curves and vertical lines—indicating the shape's edges.
• Top curve and bottom curve define the functions between which you calculate the area.
• Vertical lines indicate the limits or bounds on the x-axis, further specifying where the region starts and stops.

In the given problem, the curves \(y = x^2 + 1\) and \(y = x^2\) form the top and bottom boundaries, and the lines \(x = 1\) and \(x = 3\) contain the region horizontally. Together, these four constraints enclose an area which needs to be evaluated for size.

Integration by subtraction

Integration by subtraction is a useful technique to find the area between two curves. By subtracting, we calculate the net area encompassed between the curves over a specific interval.

This involves several clear steps:

• Determine the "top" and "bottom" functions, i.e., which function lies above the other in the graph for the given range.
• Set up the integral using the formula: \[ \int_a^b [f(x) - g(x)] \, dx \]
• Evaluate the integral for the area.

In the example problem, the top curve is \(y = x^2 + 1\) and the bottom is \(y = x^2\). By integrating:\[\int_1^3 [(x^2 + 1) - (x^2)] \, dx = \int_1^3 1 \, dx,\]we find the enclosed area to be \(2\) square units, showing how subtraction directly impacts the calculation.

Calculus problem-solving

Solving calculus problems, like finding the area between curves, systematically involves a methodical approach. Breaking down the problem into smaller, more manageable steps is crucial.

Here's a simple path to tackle such problems:

• Start by visualizing the region—draw a sketch if needed to understand the geometry of the situation.
• Clearly identify the functions involved and the region they enclose.
• Set up your equation properly, remembering the role of each function in relation to others. For bound regions, distinguish between the top and bottom curves.
• Perform integration using the correct limits.
• Verify your steps and computed values ensuring you address the specific boundaries and points.

In solving the problem of finding the area between the specified curves, these steps guide the process—helping you apply subtraction for integration, leading to correct solutions.