Problem 46 - Thomas Calculus

Question

Find the areas of the regions enclosed by the lines and curves in Exercises \(41-50 .\) $$ y=7-2 x^{2} \quad \text { and } \quad y=x^{2}+4 $$

Step-by-Step Solution

Answer

The area between the curves is 4 square units.

Step 1: Identify the Area Between Curves

To find the area between two curves, we first identify the equations of the curves: \( y = 7 - 2x^2 \) and \( y = x^2 + 4 \). The area we need is between these curves from where they intersect.

Step 2: Find Points of Intersection

Set the two equations equal to each other to find points of intersection: \( 7 - 2x^2 = x^2 + 4 \). Combine like terms to get: \( 7 - 4 = 3x^2 \), simplifying to \( 3 = 3x^2 \). Divide both sides by 3: \( x^2 = 1 \). Solving for \( x \), we get \( x = \pm 1 \).

Step 3: Solve for Intersection Points

Substitute \( x = 1 \) and \( x = -1 \) into either of the original equations to find the corresponding \( y \)-values. For \( x = 1 \), \( y = 7 - 2(1)^2 = 5 \). For \( x = -1 \), \( y = 5 \). Therefore, the points of intersection are \((1, 5)\) and \((-1, 5)\).

Step 4: Set Up the Integral for Area

The area \( A \) between the curves is found by integrating the difference of the functions from \( x = -1 \) to \( x = 1 \):\[A = \int_{-1}^{1} [(7 - 2x^2) - (x^2 + 4)] \, dx\]

Step 5: Simplify the Integrand

Simplify the expression inside the integral: \( (7 - 2x^2) - (x^2 + 4) = 7 - 2x^2 - x^2 - 4 = 3 - 3x^2 \). The integral becomes: \[A = \int_{-1}^{1} (3 - 3x^2) \, dx\]

Step 6: Compute the Definite Integral

Integrate \( 3 - 3x^2 \) term by term: - The integral of \(3\) is \(3x\).- The integral of \(-3x^2\) is \(-x^3\).Compute the definite integral from \(-1\) to \(1\):\[A = \left[ 3x - x^3 \right]_{-1}^{1} = \left[(3(1) - (1)^3) - (3(-1) - (-1)^3)\right]\]

Step 7: Evaluate the Expression

Substitute and evaluate:\((3(1) - 1^3) = (3 - 1) = 2\) \((3(-1) - (-1)^3) = (-3 + 1) = -2\)Now compute: \(2 - (-2) = 4\). The area is therefore equal to \(4\).

Key Concepts

Definite Integral

The concept of the Definite Integral is crucial for finding areas between curves. It provides a way to calculate the total accumulation of quantities. Imagine you want to find the exact space between two bounding lines of a region. The definite integral gives you a precise number for this space. By integrating over an interval, typically from point A to B, we gain a full understanding of how much space is enclosed between the functions across that range.In solving the given problem, we used the definite integral to find the space between the curves given by the equations \( y = 7 - 2x^2 \) and \( y = x^2 + 4 \). Specifically, we found the integral of the difference of these two functions over the interval determined by their points of intersection. This process is not just useful for geometry but also appears in various physical applications like computing work done or heating effects.

Points of Intersection

Finding the Points of Intersection between curves is vital because they define the limits of integration. These are the x-values where the graphs of two functions touch or cross each other, indicating where one curve drops below the other.In our exercise, we set the equations \( 7 - 2x^2 = x^2 + 4 \) equal to identify these intersection points. We then solved for \( x \) to find that the curves intersect at \( x = 1 \) and \( x = -1 \). Substituting these values back into either equation confirmed the points of intersection as \( (1, 5) \) and \( (-1, 5) \).These intersections are crucial because they tell us where to "start" and "stop" when calculating the area between the curves using definite integration.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients. They form the backbone of various problems involving curves and areas. Simple polynomial functions such as \( ax^2 + bx + c \) are often used, as they are easy to manipulate and integrate.In the exercise provided, both \( y = 7 - 2x^2 \) and \( y = x^2 + 4 \) are polynomials. Their simple structures make it feasible to determine their points of intersection algebraically and perform calculus operations like integration.Working with polynomials requires understanding their degree and behavior, such as how their graphs might uptick or downtick at certain intervals. This helps in visualizing the problem and setting up the definite integrals correctly.

Definite Integration Steps

When you compute the area between curves using integration, there are specific Definite Integration Steps to follow. First, determine the points of intersection to establish the limits of integration. Next, set up the integral expression by subtracting one polynomial from the other to form the integrand. Ensure to simplify this expression as necessary.Here's how it looks in practice:

• Identify the polynomials and their intersection points.
• Subtract the lower function from the upper function within the integration limits.
• Simplify the integrand, as seen with the integration of \(3 - 3x^2\) in our example.
• Evaluate the integral: \[ \int_{-1}^{1} (3 - 3x^2) \, dx \] results in \(3x - x^3\) after integration, evaluated from -1 to 1.
• Calculate the definite integral and interpret the result; here it equates to 4, representing the area.

Each step is critical to ensure precision and correctness in computing the desired area between the curves.