Problem 10 - Precalculus: Mathematics for Calculus

Question

In these exercises we estimate the area under the graph of a function by using rectangles. (a) Estimate the area under the graph of \(f(x)=25-x^{2}\) from \(x=0\) to \(x=5\) using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

Step-by-Step Solution

Answer

The area estimate using right endpoints is 70 (underestimate) and using left endpoints is 95 (overestimate).

Step 1: Understand the Problem

We need to estimate the area under the curve of the function \( f(x) = 25 - x^2 \) from \( x = 0 \) to \( x = 5 \) using rectangle approximation methods. Part (a) requires the use of right endpoints, while part (b) requires the use of left endpoints.

Step 2: Divide the Interval

For both parts of the problem, divide the interval \([0,5]\) into 5 equal parts. This means each subinterval has a width of \( \Delta x = 1 \). The subintervals are \([0,1], [1,2], [2,3], [3,4], [4,5]\).

Step 3: Part (a): Use Right Endpoints

Using the right endpoints of each subinterval, they will be \( x = 1, 2, 3, 4, 5 \). Calculate the height of each rectangle by evaluating the function \( f(x) \) at these right endpoints: - \( f(1) = 24 \)- \( f(2) = 21 \)- \( f(3) = 16 \)- \( f(4) = 9 \)- \( f(5) = 0 \)Calculate the area of each rectangle and sum them: \[ A = \sum_{i=1}^{5} f(x_i^R) \times \Delta x = (24 + 21 + 16 + 9 + 0) \times 1 = 70 \]

Step 4: Determine Over/Underestimate for Right Endpoints

Since the function \( f(x) = 25 - x^2 \) is decreasing over \([0,5]\), using right endpoint rectangles will result in an underestimation of the area.

Step 5: Part (b): Use Left Endpoints

Using the left endpoints of each subinterval, they will be \( x = 0, 1, 2, 3, 4 \). Calculate the height of each rectangle by evaluating the function \( f(x) \) at these left endpoints:- \( f(0) = 25 \)- \( f(1) = 24 \)- \( f(2) = 21 \)- \( f(3) = 16 \)- \( f(4) = 9 \)Calculate the area of each rectangle and sum them:\[ A = \sum_{i=0}^{4} f(x_i^L) \times \Delta x = (25 + 24 + 21 + 16 + 9) \times 1 = 95 \]

Step 6: Determine Over/Underestimate for Left Endpoints

Since the function is decreasing, using left endpoint rectangles will result in an overestimation of the area.

Key Concepts

Right Endpoint Approximation

The right endpoint approximation method is used to estimate the area beneath a curve by taking the rightmost value of each subinterval to determine the height of the rectangle. This means for the interval \[0,5\], each subinterval, such as \[0,1\] or \[1,2\], uses the value at the right end point (ex: f(1), f(2), etc.) to calculate the rectangle's height.

• Step 1: Divide the interval into equal parts.
• Step 2: For each rectangle, measure the height using the function value at the right endpoint of each subinterval.
• Step 3: Compute the area by summing the area of all rectangles, where each rectangle's area = height \( \times \) width.

This approach is particularly useful when you want to get an approximation without complicated calculations. The number of subintervals directly affects the accuracy of the approximation. The more subintervals, the closer the approximation is to the actual area.

Left Endpoint Approximation

Unlike the right endpoint method, the left endpoint approximation uses the leftmost point of each subinterval to determine the height of the rectangle. In this method, for the interval \[0,5\], each subinterval uses the function value at the beginning of the interval, such as f(0), f(1), etc.

• Step 1: Divide the interval into equal sections.
• Step 2: Use the function value at the start of each subinterval to define the height of each rectangle.
• Step 3: Compute the total area by adding the areas of all rectangles.

This method helps in quickly assessing the area while understanding how the function behaves at each point in the interval. Again, the more subintervals you divide the main interval into, the better your estimate tends to be.

Underestimate

An underestimate in area approximation occurs when the estimated area is less than the actual area under the curve. This typically happens with decreasing functions when using the right endpoint method.

• Since the curve is moving downwards, each rectangle, drawn using the right endpoint, falls short of capturing the full area under the curve for each subinterval.
• This leads to a scenario where the sum of all rectangle areas falls below the true area bounded by the curve and the x-axis.

Thus, the right endpoint approximation results in a less than accurate representation unless adjustments are made, such as increasing the number of subintervals.

Overestimate

An overestimate happens when the calculated area exceeds the actual area under the curve. This often occurs with functions that are decreasing and when the left endpoint approximation method is employed.

• Due to the decreasing nature of the function, using the left endpoint includes portions of the area that aren't under the curve, thereby inflating the total calculated area.
• The rectangles stretch above the curve, especially noticeable when there are fewer subintervals, making the estimation less accurate.

The solution to reducing the error in an overestimate involves using more subintervals for better accuracy and ensuring a closer fit to the actual curve.