Problem 26 - Algebra 2

Question

Graph each curve. Use inscribed rectangles to approximate the area under the curve for the interval and rectangle width given. $$ y=x^{2}, 3 \leq x \leq 5,0.5 $$

Step-by-Step Solution

Answer

The short answer to this problem is to graph the curve and use inscribed rectangles to approximate the area under the curve on the given interval. The sum of areas of these rectangles will give an approximation of the area under the curve.

Step 1: Draw the Graph

First, you need to draw the graph of the function \( y = x^{2} \). This is a simple parabola opening upwards. It's always helpful to visualize the function graphically.

Step 2: Identify the Interval

Identify the interval from \( x = 3 \) to \( x = 5 \) on the graph of \( y = x^{2} \). This specific region under the curve is the area we're going to approximate.

Step 3: Draw the Inscribed Rectangles

Draw four vertical rectangles (since the width is 0.5) inside the given interval of the curve, and make sure they do not exceed the curve. Since they are inscribed rectangles, they should fit perfectly within the curve.

Step 4: Calculate the Area of Each Rectangle

Calculate the area of each rectangle. The area of a rectangle is given by \( base \times height \). Here, the base is the width of the rectangle which is 0.5. The height will be the value of the function \( y = x^{2} \) at the left end point of each interval. For instance, for the first interval between 3 and 3.5, the height will be \( y = (3)^{2} = 9 \). So, the area of the first rectangle is \( 9 \times 0.5 = 4.5 \). Perform similar calculations for the other rectangles.

Step 5: Add up the Areas

Add up the areas of the four rectangles to get an approximation for the area under the curve. This sum of areas will be an underestimate of the actual area under the curve, but it can give a rough approximation.

Key Concepts

Inscribed Rectangles

When approximating the area under a curve, one common technique is to use inscribed rectangles. In this context, an inscribed rectangle is a rectangle that fits snugly under the curve, without surpassing it at any point. These rectangles help approximate the area by filling the space beneath the curve in small, manageable sections.

To use inscribed rectangles:

• Divide the interval of interest into smaller sub-intervals. The width of each rectangle corresponds to these sub-intervals.
• The height of each rectangle is determined by the function's value at a chosen point within the sub-interval, usually the left endpoint when doing a left Riemann sum.
• Sum the areas of all the rectangles to approximate the total area under the curve.

Since the inscribed rectangles do not fully cover the area under the curve but sit inside of it, the total area sum will always be an underestimate of the actual curve area.

Parabolas

Parabolas are symmetrical, U-shaped curves that can open upwards or downwards. The function given as an example, \( y = x^2 \), represents a parabola that opens upwards. This is a basic form of a quadratic equation, where the vertex of the parabola is its lowest point depending on the coefficient in terms of growth or decay.

Some properties of parabolas:

• The vertex of the parabola \( y = x^2 \) is at the origin (0,0), but we focus on a segment starting from \( x = 3 \) in our interval.
• As \( x \) increases or decreases from the vertex, \( y = x^2 \) increases as well, exhibiting a squared growth pattern.
• The axis of symmetry is a vertical line that passes through the vertex, ensuring that the parabola is symmetrical on either side of this line.

Understanding parabolas and their characteristics helps in predicting how they behave in a given interval and in estimating areas.

Graphing Functions

Graphing functions is crucial for visualizing mathematical concepts, especially when estimating areas under curves. With graphing, you can see the shape and characteristics of functions like parabolas, linear graphs, and waves.

Steps to graph a function:

• Identify the function you need to graph. For \( y = x^2 \), it will form a parabola.
• Determine key points across the given interval, such as the endpoints and any vertex or turning points.
• Plot these points on a coordinate plane and connect them smoothly to reveal the full shape of the function.
• Ensure your graph accurately represents any specific characteristics like symmetry, growth, or decay rates based on the function's equation.

Graphing not only aids in understanding but also serves as a visual guide when calculating complex areas or geometries.

Area Under the Curve

The concept of the area under a curve is fundamental in calculus, representing the total space enclosed between a graph and the \( x \)-axis over a given interval. In calculus, integrating a function over an interval is the primary method to find this area exactly but sometimes an approximate area is useful for estimation purposes.

To approximate the area under a curve:

• Pick a method like inscribed rectangles or trapezoidal approximations, depending on the desired accuracy and simplicity.
• Divide the interval into smaller sections and calculate the appropriate heights based on the function value at specific points.
• Calculate and sum the individual areas of these basic shapes to approximate the total area under the curve.

In approximations like with inscribed rectangles, the sum provides an understanding of how much space the curve occupies in the specified interval, which can be particularly useful in scenarios where exact calculations are complex or unnecessary.