Estimating Area with Finite Sums: Videos & Practice Problems

To estimate the area under a curve, we can divide the region into rectangles, using left, right, or midpoint approximations. The width of each rectangle, denoted as $\Delta x$, is calculated as $(b-a)/n$. The height is determined by the function value at the chosen endpoints. More rectangles yield a more accurate approximation of the area, enhancing our understanding of integral calculus.

Estimating the Area Under a Curve Using Left Endpoints

Understanding how to find the area underneath a curve is a fundamental concept in calculus, particularly when dealing with functions that do not form simple geometric shapes like triangles or squares. To estimate this area, we can use a method that involves dividing the area under the curve into smaller, manageable sections, typically rectangles.

The first step in this process is to determine the number of rectangles, denoted as n, which also represents the number of subintervals. By breaking the curve into these rectangles, we can approximate the area by summing the areas of each rectangle. The area of a rectangle is calculated using the formula:

Area = Width × Height

In this context, the width of each rectangle is referred to as Δx, which can be calculated using the formula:

Δx = (b - a) / n

Here, a and b are the lower and upper bounds of the interval over which we are calculating the area. For example, if we are estimating the area under a curve from x = 0 to x = 2 using four rectangles, we would calculate:

Δx = (2 - 0) / 4 = 0.5

Next, to find the height of each rectangle, we evaluate the function at specific points known as the left endpoints of each rectangle. For instance, if we have four rectangles, we would evaluate the function at f(0), f(0.5), f(1), and f(1.5). The heights of the rectangles are then determined by these function values.

Once we have the width and heights, we can compute the total area under the curve by summing the areas of all rectangles:

Total Area ≈ Δx × (f(0) + f(0.5) + f(1) + f(1.5))

As we increase the number of rectangles, the approximation becomes more accurate. For example, using two rectangles might yield an area of 7, while using four rectangles could give an area of 6.25. The more rectangles we use, the closer our estimate will be to the actual area under the curve, as the rectangles will better conform to the shape of the curve, reducing the area that extends beyond the curve.

In summary, estimating the area under a curve involves dividing the area into rectangles, calculating their widths and heights based on the function values, and summing their areas. This method provides a practical approach to approximating areas that cannot be easily calculated using standard geometric formulas.

Use five rectangles to estimate the area under the curve of $f\left(x\right)=x^2$ from $x=0$ to $x=5$ using left endpoints.

• A. $A=30$
• B. $A=55$
• C. $A=41.25$
• D. $A=41.67$

Answer: (A)

Use two rectangles to estimate the area under the curve of $f\left(x\right)=\frac12x^2$ from $x=0$ to $x=3$ using left endpoints.

• A. $A=1.6875$
• B. $A=4.21875$
• C. $A=8.4375$
• D. $A=4.5$

Answer: (A)

Use four rectangles to estimate the area under the curve of $f\left(x\right)=x^2+2$ from $x=0$ to $x=2$ using left endpoints.

• A. $A=5.00$
• B. $A=6.67$
• C. $A=5.75$
• D. $A=7.75$

Answer: (C)

Estimating the Area Under a Curve with Right Endpoints & Midpoint

Estimating the area under a curve can be effectively achieved by dividing the area into rectangles, a method known as Riemann sums. This approach involves using subintervals, where the width of each rectangle remains constant, while the height varies based on the chosen points along the curve. The left endpoint approximation, where the height of each rectangle is determined by the function value at the left side of the interval, often leads to an overestimation of the area.

To enhance accuracy, other methods can be employed, such as right endpoint and midpoint approximations. In the right endpoint method, the height of each rectangle is determined by the function value at the right side of the interval. For example, if the width of each rectangle is denoted as \( \Delta x \), the area of a rectangle can be calculated using the formula:

Area = \( \Delta x \times f(x_{\text{right}}) \)

where \( x_{\text{right}} \) is the right endpoint of the interval. This method may yield different area estimates compared to the left endpoint method, as it focuses on the rightmost values of the function.

The midpoint approximation offers a balanced approach, often resulting in a more accurate estimate. In this method, the height of each rectangle is determined by the function value at the midpoint of each interval. To find the midpoint, one can average the left and right endpoints of the interval. For instance, the midpoint between 0 and 1 is:

Midpoint = \( \frac{0 + 1}{2} = 0.5 \

Thus, the area for the midpoint approximation can be calculated as:

Area = \( \Delta x \times f\left(\frac{x_{\text{left}} + x_{\text{right}}}{2}\right) \)

In this case, the function values at the midpoints provide a more balanced estimation of the area, as they account for both the overestimations and underestimations that occur with the left and right endpoint methods.

Throughout these methods, the width of the rectangles remains constant unless the number of rectangles or the interval changes. The key takeaway is that while these approximations provide useful estimates for the area under a curve, they do not yield the exact area, which can only be determined through integration.

Practicing these methods will enhance your understanding of how to estimate areas under curves effectively, preparing you for more complex applications in calculus.

Use three rectangles to estimate the area under the curve of $f\left(x\right)=\frac13x^3$ from $x=0$ to $x=3$ using the right endpoints.

• A. $A=3$
• B. $A=12$
• C. $A=6.5$
• D. $A=5.3$

Answer: (B)

Use three rectangles to approximate the area under the curve of $f\left(x\right)=3\left(x-2\right)^2$ from $x=0$ to $x=3$ using the midpoint rule.

• A. $A=9.00$
• B. $A=6.00$
• C. $A=8.25$
• D. $A=15.0$

Answer: (C)

Estimating the Area Under a Curve with Right Endpoints & Midpoint Example 1

To approximate the area under the curve of the function \( f(x) = x^3 \) on the interval from 0 to 3 using three rectangles, we can employ three different methods: left endpoints, right endpoints, and midpoints. Each method provides a different approximation of the area, highlighting the importance of the chosen points for calculating the heights of the rectangles.

First, we calculate the width of each rectangle, denoted as \( \Delta x \). This is determined by the formula:

\( \Delta x = \frac{b - a}{n} \)

where \( b \) is the upper limit (3), \( a \) is the lower limit (0), and \( n \) is the number of rectangles (3). Thus, we have:

\( \Delta x = \frac{3 - 0}{3} = 1 \)

This width remains constant across all three methods.

For the left endpoint approximation, we calculate the area as follows:

Area \( A_L = \Delta x \cdot (f(0) + f(1) + f(2)) \)

Calculating the function values:

• \( f(0) = 0^3 = 0 \)
• \( f(1) = 1^3 = 1 \)
• \( f(2) = 2^3 = 8 \)

Thus, the area becomes:

Area \( A_L = 1 \cdot (0 + 1 + 8) = 9 \

Next, for the right endpoint approximation, we adjust our calculations to use the rightmost points:

Area \( A_R = \Delta x \cdot (f(1) + f(2) + f(3)) \)

Calculating the function values:

• \( f(1) = 1^3 = 1 \)
• \( f(2) = 2^3 = 8 \)
• \( f(3) = 3^3 = 27 \)

Thus, the area becomes:

Area \( A_R = 1 \cdot (1 + 8 + 27) = 36 \)

Finally, for the midpoint approximation, we find the midpoints of each interval:

• Midpoint between 0 and 1: \( 0.5 \)
• Midpoint between 1 and 2: \( 1.5 \)
• Midpoint between 2 and 3: \( 2.5 \)

The area is calculated as:

Area \( A_M = \Delta x \cdot (f(0.5) + f(1.5) + f(2.5)) \)

Calculating the function values:

• \( f(0.5) = (0.5)^3 = 0.125 \)
• \( f(1.5) = (1.5)^3 = 3.375 \)
• \( f(2.5) = (2.5)^3 = 15.625 \)

Thus, the area becomes:

Area \( A_M = 1 \cdot (0.125 + 3.375 + 15.625) = 19.125 \)

In summary, the approximations yield different results: the left endpoint gives an area of 9, the right endpoint gives 36, and the midpoint gives 19.125. These variations illustrate how the choice of endpoints affects the accuracy of area approximations under a curve.

Here’s what students ask on this topic:

What is the process for estimating the area under a curve using rectangles?

To estimate the area under a curve, we divide the region into rectangles. The process involves selecting a number of rectangles, denoted as $n_{}$, and calculating the width of each rectangle as $\frac{b-a}{n}$. The height of each rectangle is determined by the function value at a chosen endpoint, which can be the left, right, or midpoint of the interval. The area of each rectangle is then calculated as width times height, and the sum of these areas provides an approximation of the total area under the curve. Increasing the number of rectangles generally improves the accuracy of the approximation.

How do left, right, and midpoint approximations differ in estimating area under a curve?

Left, right, and midpoint approximations differ in how they determine the height of each rectangle used to estimate the area under a curve. In a left endpoint approximation, the height is determined by the function value at the left side of each subinterval. In a right endpoint approximation, the height is determined by the function value at the right side. The midpoint approximation uses the function value at the midpoint of each subinterval. These differences affect the accuracy of the estimation, with midpoint approximations often providing a more balanced estimate by averaging overestimations and underestimations.

Why does increasing the number of rectangles improve the accuracy of area estimation under a curve?

Increasing the number of rectangles improves the accuracy of area estimation under a curve because it reduces the size of each rectangle, allowing them to better conform to the shape of the curve. Smaller rectangles can more closely follow the curve's fluctuations, minimizing the overestimation or underestimation that occurs when using larger rectangles. As the number of rectangles approaches infinity, the approximation becomes more precise, converging to the exact area under the curve, which is the integral of the function over the interval.

What is the formula for calculating the width of rectangles in area estimation?

The formula for calculating the width of rectangles in area estimation is given by $\frac{b-a}{n}$, where $a$ and $b$ are the endpoints of the interval over which the area is being estimated, and $n$ is the number of rectangles or subintervals. This formula ensures that each rectangle has an equal width, allowing for a consistent approach to estimating the area under the curve.

How do you calculate the height of rectangles in area estimation using function values?

The height of rectangles in area estimation is calculated using the function values at specific points within each subinterval. For a left endpoint approximation, the height is the function value at the left endpoint of the subinterval. For a right endpoint approximation, it is the function value at the right endpoint. In a midpoint approximation, the height is the function value at the midpoint of the subinterval. These function values are used to determine the height of each rectangle, which is then multiplied by the width to calculate the area of each rectangle.