Problem 28 - Algebra 2

Question

Graph each curve. Use inscribed rectangles to approximate the area under the curve for the interval and rectangle width given. $$ y=x^{3}, 1 \leq x \leq 3,0.25 $$

Step-by-Step Solution

Answer

By calculating the area of each inscribed rectangle and summing them up, the approximate area under the curve of the function \( y = x^3 \) on the interval [1, 3] is found.

Step 1: Graph the Function

First, plot the curve of the function \( y = x^3 \) in the given interval \([1, 3]\).

Step 2: Calculate the Height of Rectangles

For inscribed rectangles, the height of each rectangle will be determined based on the function value at the left endpoint of the corresponding subinterval. Our interval \([1, 3]\) is divided into 8 subintervals: \([1, 1.25]\), \([1.25, 1.5]\), \([1.5, 1.75]\), \([1.75, 2]\), \([2, 2.25]\), \([2.25, 2.5]\), \([2.5, 2.75]\), and \([2.75, 3]\). Thus, for each rectangle, the height corresponds to \(x^3\) where \(x\) is the left endpoint of the subinterval.

Step 3: Calculate the Area of Rectangles

The area of each rectangle is calculated by multiplying the width (which is 0.25) by the height (corresponds to \(x^3\) where \(x\) is the left endpoint of the subinterval).

Step 4: Compute the Total Sum

After determining the area of each rectangle, sum up these areas to obtain the approximation of the area under the curve.

Key Concepts

Inscribed Rectangles

Inscribed rectangles are a helpful tool when approximating the area under a curve, particularly when working with functions on a specified interval. They are rectangles drawn inside a curve such that their tops touch the curve from below.

• The width of each rectangle is determined by dividing the interval into equal parts.
• The height is the function value at the left endpoint of each subinterval.

By choosing the left endpoint, the rectangles stay completely within the boundaries of the curve, ensuring an underestimate of the actual area. This method sets the foundation for calculating Riemann sums and understanding integral calculus. The degree of approximation improves as you increase the number of rectangles.

Graphing Polynomial Functions

Graphing polynomial functions like the cubic function, in this example, requires understanding their general shape and behavior. The function given is \(y = x^3\).

• Generally, cubic functions have an S-shaped curve which passes through the origin if not shifted vertically or horizontally.
• They increase to infinity as \(x\) approaches positive infinity and decrease to negative infinity as \(x\) approaches negative infinity.

For the interval \([1, 3]\), you only look at a portion of this curve. Plotting it requires knowing critical points and how the polynomial behaves on this specific segment. By accurately sketching the function, you can properly align your inscribed rectangles.

Riemann Sum

The Riemann sum is a core principle in calculus for estimating the area under a curve. It involves summing up areas of multiple rectangles to derive an approximation.

• Each rectangle's area is determined by multiplying its width by its height.
• The height of each rectangle is the function's value at a given point within the subinterval (here, the left endpoint).

In this problem, the interval \([1, 3]\) is divided into 8 rectangles of equal width 0.25. Calculating each individual rectangle's area and summing them provides an approximation of the entire area under the curve. The more rectangles used, the closer the approximation will be to the true value.

Area under the Curve

The area under the curve is a fundamental concept in calculus, revealing the total value accumulated by the function between two points. Estimating this area requires breaking the curve into simpler, manageable shapes, such as rectangles.

• This area represents the integral of the function over the given interval.
• Approximations can be improved by increasing the number of rectangles or moving towards numerical integration methods.

In this example, using inscribed rectangles provides a practical way to approximate the integral of \(y = x^3\) from 1 to 3. Understanding this concept helps build intuition about integration and its applications in real-world problems.