Problem 25 - Algebra 2

Question

Graph each curve. Use inscribed rectangles to approximate the area under the curve for the interval and rectangle width given. $$ y=3 x^{2}+2,2 \leq x \leq 4,1 $$

Step-by-Step Solution

Answer

The approximate area under the curve \(y=3x^2+2\) from \(x=2\) to \(x=4\) using one inscribed rectangle is 28 square units.

Step 1: Graph the Function

First, the function \(y=3x^2+2\) must be graphed. This function represents a parabola which opens upwards with a y-intercept of 2. Hence, the parabola will be above the x-axis for the given interval \([2,4]\).

Step 2: Identify Intervals and Create Rectangles

The interval from 2 to 4 is broken into subintervals, each representing the width of an inscribed rectangle under the curve. For this example, assume there is 1 rectangle (for simplicity). So, the width is (4-2)=2 units. The height of the rectangle, \(h\), is determined by the function value at the left endpoint of the interval, i.e., \(h=f(2)=3(2)^2+2=14\ units. This rectangle is inscribed under the curve.

Step 3: Calculate Area

Now, the area of this rectangle can be calculated to approximate the area under the curve for the given interval. It is computed using the formula for the area of a rectangle, \(A = width \times height\). Therefore, \(A = 2 \times 14 = 28\ square\ units.\) This is the approximate area under the curve between \(x=2\) and \(x=4\).

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that can be represented by a quadratic equation of the form \(y = ax^2 + bx + c\). In this particular example, we have the function \(y = 3x^2 + 2\). This formula shows a parabola that opens upwards, as the coefficient of \(x^2\) is positive. The parabola's vertex serves as either the lowest point or the highest point of the curve depending on the direction it opens. For upward-opening parabolas, like ours, the vertex is the lowest point. However, since our focus is only on the interval \([2,4]\), we don't need to compute the vertex for this exercise.

• The coefficient \(3\) in \(3x^2\) determines the steepness of the curve: the larger the value, the steeper the parabola.
• The constant term \(2\) tells us that the parabola crosses the y-axis at \(y=2\), this is the y-intercept.

Understanding how to graph and interpret parabolic functions is crucial for visualizing how the curve behaves over a given interval.

Rectangular Approximation Method

The Rectangular Approximation Method is a numerical technique used to estimate the area under a curve. This method involves filling the space under the curve with rectangles, whose areas are simple to calculate.

• Rectangles are typically inscribed at specific x-values within the interval you are evaluating, like the left endpoint, right endpoint, or the midpoint.
• The more rectangles you use, generally, the more accurate your approximation will be.

In this problem, only a single rectangle is used to approximate the area under the parabola from \(x=2\) to \(x=4\). The method involves calculating the height of the rectangle using the function value at the left endpoint of the interval. Calculating this gives us the height: \(h = 3(2)^2 + 2 = 14\). Multiplying this height by the width of 2 (difference between the x-values), we obtain the area approximation of \(28\) square units.As we are using only one rectangle, the approximation might not be perfectly accurate, but it demonstrates the concept of using rectangular approximation to estimate areas.

Inscribed Rectangles

When approximating areas under curves using the Rectangular Approximation Method, the concept of inscribed rectangles is essential. Inscribed rectangles are those that lie entirely below the curve, with their top edges touching the curve at one point, typically at a defined endpoint. Here is how it works:

• Height: The height of each rectangle is determined by the value of the function at a specific point within each subinterval. In this exercise, it is the left endpoint, so the height is \(f(2)\).
• Width: The width in our problem is the length of the interval divided by the number of rectangles, which equals \(2\) since we are using one rectangle.

The choice of endpoints - whether using the left, right, or midpoint - affects how these rectangles fit under the curve and thus, influences the accuracy of the approximation. In our exercise, using a single rectangle fits the technique conceptually without overwhelming complexity. The entire rectangle lies beneath the parabola, making it inscribed.