Estimating Area with Finite Sums: Videos & Practice Problems

To estimate the area under a curve, we can divide it into rectangles, using left, right, or midpoint approximations. The width of each rectangle, denoted as Δx, is calculated using the formula $(b-a)/n$. The height is determined by the function value at the chosen endpoint. Increasing the number of rectangles improves accuracy, reducing overestimation or underestimation of the area, which is crucial for understanding concepts like consumer surplus and market efficiency.

Estimating the Area Under a Curve Using Left Endpoints

Understanding how to find the area underneath the curve of a function is a fundamental concept in calculus. This process involves estimating the area by dividing the region under the curve into smaller, manageable shapes, typically rectangles. The number of rectangles used for this approximation is denoted by n, which also represents the number of subintervals.

To estimate the area under a curve, one common method is to use left endpoints of the rectangles. For instance, if we want to approximate the area under a curve from a to b, we can calculate the width of each rectangle using the formula:

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

Here, b is the upper limit, a is the lower limit, and n is the number of rectangles. The height of each rectangle is determined by the function value at the left endpoint of each subinterval.

For example, if we have a function and we choose to use two rectangles, we would calculate the area of each rectangle as:

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

where \(f(x_i)\) is the function value at the left endpoint of the rectangle. By summing the areas of all rectangles, we can obtain an approximate area under the curve.

As we increase the number of rectangles, the approximation becomes more accurate. For instance, using four rectangles instead of two can yield a different result, which may be closer to the actual area. This is because more rectangles can better conform to the shape of the curve, reducing the area that extends beyond the curve.

In summary, the process of 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. The more rectangles used, the more accurate the approximation will be, making this a crucial technique in integral calculus.

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 $f\left(x\right)=\frac{1}{2}{x}^2$ 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:

\[ \text{Area} = \Delta x \cdot f(x_{\text{right}}) \]

where \( x_{\text{right}} \) represents the right endpoint of the interval. This method may yield different area estimates compared to the left endpoint method due to the varying heights of the rectangles.

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:

\[ x_{\text{mid}} = \frac{0 + 1}{2} = 0.5 \]

Thus, the area for the rectangle using the midpoint approximation can be expressed as:

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

In this case, the height is determined by evaluating the function at the midpoint, which often provides a more balanced estimation of the area under the curve. The process remains consistent across all methods, with the primary difference being the selection of the x-values that determine the heights of the rectangles.

Ultimately, while these methods provide useful approximations, they do not yield the exact area under the curve. The accuracy of the estimates can improve by increasing the number of rectangles, thereby reducing the width of each rectangle, which allows for a finer approximation of the curve's area. Engaging with practice problems will further solidify understanding of these concepts and their applications in estimating areas under curves.

Use three rectangles to estimate the area under the curve of $f\left(x\right)=\frac{1}{3}{x}^3$ 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 $f\left(x\right)=3{(x-2)}^2$ 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 \( f(x) = x^3 \) over the interval from 0 to 3 using three rectangles, we can utilize three different methods: left endpoints, right endpoints, and midpoints. Each method provides a different approximation based on how we calculate 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 methods since the interval and the number of rectangles do not change.

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

\[A_L = \Delta x \cdot (f(0) + f(1) + f(2))\]Calculating the function values:- \( f(0) = 0^3 = 0 \)- \( f(1) = 1^3 = 1 \)- \( f(2) = 2^3 = 8 \)Thus, the area becomes:\[A_L = 1 \cdot (0 + 1 + 8) = 9\]

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

\[A_R = \Delta x \cdot (f(1) + f(2) + f(3))\]Calculating the function values:- \( f(1) = 1^3 = 1 \)- \( f(2) = 2^3 = 8 \)- \( f(3) = 3^3 = 27 \)Thus, the area becomes:\[A_R = 1 \cdot (1 + 8 + 27) = 36\]

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

- The midpoint between 0 and 1 is \( 0.5 \)- The midpoint between 1 and 2 is \( 1.5 \)- The midpoint between 2 and 3 is \( 2.5 \)The area is calculated as:\[A_M = \Delta x \cdot (f(0.5) + f(1.5) + f(2.5))\]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:\[A_M = 1 \cdot (0.125 + 3.375 + 15.625) = 19.125\]

In summary, the approximated areas under the curve using the three methods are:

• Left endpoint approximation: \( 9 \)
• Right endpoint approximation: \( 36 \)
• Midpoint approximation: \( 19.125 \)

These results illustrate how different methods yield varying approximations of the area under the curve, with the left endpoint providing an underestimation, the right endpoint an overestimation, and the midpoint offering a value that lies between the two.

Here’s what students ask on this topic:

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

Estimating the area under a curve using rectangles involves dividing the region into smaller sections, typically rectangles, and summing their areas. This method is known as the Riemann sum. The width of each rectangle, denoted as Δx, is calculated as (b-a)/n, where b and a are the interval endpoints, and n is the number of rectangles. The height of each rectangle is determined by the function value at a specific point, which can be the left endpoint, right endpoint, or midpoint of the interval. The choice of endpoint affects the accuracy of the estimation, with midpoints often yielding the most precise results. This technique is fundamental in calculus for approximating integrals and analyzing functions.

How do left endpoint, right endpoint, and midpoint approximations differ in estimating area?

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

Why is the midpoint approximation often more accurate than left or right endpoint approximations?

The midpoint approximation is often more accurate than left or right endpoint approximations because it balances the overestimations and underestimations that occur with the other methods. By using the function value at the midpoint of each subinterval, the midpoint approximation tends to average out the areas that peek over and under the curve, leading to a more precise estimate of the true area. This method reduces the error associated with the endpoints, providing a closer approximation to the actual integral of the function over the interval.

How does increasing the number of rectangles affect the accuracy of area estimation under a curve?

Increasing the number of rectangles in the estimation process generally improves the accuracy of the area estimation under a curve. As the number of rectangles increases, the width of each rectangle (Δx) decreases, allowing the rectangles to better conform to the shape of the curve. This reduces the amount of area that either peeks over or under the curve, leading to a more precise approximation of the true area. In the limit, as the number of rectangles approaches infinity, the Riemann sum converges to the exact value of the definite 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 Δx = (b-a)/n, where b and a 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 uniform division of the interval. The width Δx is crucial for determining the dimensions of each rectangle and subsequently calculating their areas to approximate the total area under the curve.