Problem 34 - Calculus: Graphical, Numerical, Algebraic

Question

Finding Area Find the area of the region enclosed by the \(y\) -axis and the curves \(y=x^{2}\) and \(y=\left(x^{2}+x+1\right) e^{-x}\)

Step-by-Step Solution

Answer

The steps mentioned provide a guide for solving the problem. Actual solution would require numerical integration as the points of intersection of the curves and the integrals may not have standard algebraic solutions.

Step 1: Find the Intersection Points

Set the two equations equal to each other: \(x^{2}=\left(x^{2}+x+1\right) e^{-x}\). Solve this equation for \(x\). These solutions represent the points where the two curves intersect.

Step 2: Setup the Integral

The area of the region between the two curves from \(x=a\) to \(x=b\) is equal to \(\int_{a}^{b\}| f(x) - g(x) | dx\), where \(f(x)\) and \(g(x)\) are the two functions, and \(a\) and \(b\) are the intersection points found in Step 1.

Step 3: Evaluate the Integral

Compute the integral \(\int_{a}^{b\}| f(x) - g(x) | dx\). Be mindful of the absolute value, which means that the integrand will need to be split into separate intervals where one function is greater than the other.

Step 4: Adding the Areas

After computing the areas under the curves for all intervals, add them together to get the total area between the curves.

Key Concepts

Integration

Integration is a fundamental concept in calculus that helps us find the area under a curve. When dealing with the area between curves, integration allows us to accurately calculate the space enclosed by these curves. By integrating the difference between two functions, we determine the area. In simple terms:

• Think of integration as adding up tiny slices of area under a curve.
• It's like assembling a series of rectangles to fill the space beneath a curve.

Knowing how to set up and solve an integral is essential for finding areas between curves or under a single curve.

Definite Integral

A definite integral is used to find the exact area under a curve from one point to another on the x-axis. This is different from an indefinite integral, which finds a general form of an antiderivative without specific bounds.

• The definite integral is written as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.
• In this exercise, the definite integral helps compute the area enclosed by the curves between specific intersection points.

When evaluating, ensure the integral accounts for all intervals where one curve is above the other to accurately capture the area.

Intersection Points

Intersection points are where two or more curves meet. Finding these points is crucial as they define the limits of our integration when calculating the area between curves.

• Set the two functions equal to each other to find these points: \( y=x^2 \) and \( y=(x^2+x+1) e^{-x} \).
• Solving this equation gives the values of \(x\) where both curves intersect.

These points also determine the boundaries \(a\) and \(b\) in the definite integral, giving us the precise area under consideration.

Absolute Value Function

The absolute value function modifies how we calculate the area between two curves, especially when the curves cross over each other.

• The formula \( \int_{a}^{b} | f(x) - g(x) | \, dx \) uses the absolute value to ensure we always measure the upper curve minus the lower one.
• This prevents subtracting in the wrong order, which would lead to negative areas.

Imagine flipping any negative area to positive by taking the absolute value, effectively giving us the correct total area without sign errors. This ensures all areas are added correctly.