Area Under a Curve: Videos & Practice Problems

To find the area under a function, especially linear functions, we can use familiar shapes like triangles and rectangles. For curvy functions, we approximate the area by dividing it into rectangles, calculating their base and height, and summing their areas. The exact area can be determined using limits, where the number of rectangles approaches infinity, allowing for precise calculations. Key equations include $\delta x=\frac{b-a}{n}$ and the limit definition for area.

Area Under Linear Functions

Understanding how to find the area underneath a function is a crucial skill in mathematics, particularly when dealing with linear functions. The process is simplified by recognizing that the area can often be represented using familiar geometric shapes, such as triangles and rectangles.

For instance, consider the linear function \( f(x) = x \). To find the area under this curve from \( x = 0 \) to \( x = 5 \), we can visualize the area as a triangle. The base of the triangle extends from \( 0 \) to \( 5 \) on the x-axis, giving it a length of \( 5 \). The height of the triangle, which corresponds to the value of the function at \( x = 5 \), is also \( 5 \). The area \( A \) of a triangle is calculated using the formula:

\( A = \frac{1}{2} \times \text{base} \times \text{height} \)

Substituting the values, we find:

\( A = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} = 12.5 \)

This result represents the area under the curve between the function and the x-axis.

In more complex scenarios, the area under a curve may require the combination of multiple shapes. For example, to find the area under a function from \( x = -8 \) to \( x = 4 \), we can divide the area into a triangle and a rectangle. The triangle, with a base of \( 8 \) (from \( -8 \) to \( 0 \)) and a height of \( 8 \), gives an area of:

\( A_{\text{triangle}} = \frac{1}{2} \times 8 \times 8 = 32 \)

The rectangle, extending from \( 0 \) to \( 4 \) with a height of \( 8 \), has an area of:

\( A_{\text{rectangle}} = 4 \times 8 = 32 \)

Adding these areas together provides the total area under the curve:

\( A_{\text{total}} = 32 + 32 = 64 \)

However, when the function dips below the x-axis, the area calculated can yield a negative value. For example, if we consider the area from \( x = -6 \) to \( x = 8 \), the area above the x-axis is calculated as a rectangle. The base of this rectangle is \( 14 \) (from \( -6 \) to \( 0 \) and \( 0 \) to \( 8 \)), and the height is \( -6 \) (since it extends below the x-axis). Thus, the area is:

\( A = 14 \times (-6) = -84 \)

This negative area indicates that the function is below the x-axis, illustrating that the area under the curve can indeed be negative when the function dips below the x-axis. Understanding these concepts allows for a comprehensive approach to calculating areas under linear functions, whether they are above or below the x-axis.

Find the area under the curve in the function below from $x=-4$to $x=8$.

• A. 16
• B. 32
• C. 24
• D. 40

Answer: (C)

Find the area under the curve in the function below from $x=-2$to $x=8$.

• A. 34
• B. 38
• C. 50
• D. 42

Answer: (A)

Approximating Area Under the Curve

To calculate the area under a curvy function, one effective method is to approximate the area using rectangles. This approach is particularly useful when dealing with functions that do not conform to simple geometric shapes like triangles or rectangles. The fundamental idea is to divide the area under the curve into smaller rectangles, allowing for a more manageable calculation of the total area.

The area of a rectangle is determined by the formula:

Area = Base × Height

To apply this method, first identify the interval over which you want to calculate the area. For example, consider the function \( f(x) = x^2 \) over the interval from \( x = 0 \) to \( x = 5 \). If you decide to use 5 rectangles for approximation, the width (base) of each rectangle can be calculated as:

Base = (b - a) / n

where \( b \) is the upper limit of the interval, \( a \) is the lower limit, and \( n \) is the number of rectangles. In this case:

Base = (5 - 0) / 5 = 1

Next, you can choose to use either left or right endpoints to determine the height of each rectangle. For left endpoint approximation, the height of each rectangle is determined by the function value at the left endpoint of each subinterval. Conversely, for right endpoint approximation, the height is determined by the function value at the right endpoint.

For the left endpoint approximation, the heights of the rectangles would be calculated as follows:

• 1st rectangle: Height = \( f(0) = 0^2 = 0 \)
• 2nd rectangle: Height = \( f(1) = 1^2 = 1 \)
• 3rd rectangle: Height = \( f(2) = 2^2 = 4 \)
• 4th rectangle: Height = \( f(3) = 3^2 = 9 \)
• 5th rectangle: Height = \( f(4) = 4^2 = 16 \)

Calculating the area for each rectangle gives:

Area = 1 × (0 + 1 + 4 + 9 + 16) = 30

For the right endpoint approximation, the heights would be:

• 1st rectangle: Height = \( f(1) = 1^2 = 1 \)
• 2nd rectangle: Height = \( f(2) = 2^2 = 4 \)
• 3rd rectangle: Height = \( f(3) = 3^2 = 9 \)
• 4th rectangle: Height = \( f(4) = 4^2 = 16 \)
• 5th rectangle: Height = \( f(5) = 5^2 = 25 \)

Thus, the area for the right endpoint approximation is:

Area = 1 × (1 + 4 + 9 + 16 + 25) = 55

These two methods yield different approximations for the area under the curve, with the left endpoint providing an underestimation and the right endpoint providing an overestimation. This discrepancy arises because the rectangles may not perfectly align with the curve, leading to areas that are either included or excluded. To improve accuracy, one might average the two results or use more rectangles to refine the approximation.

This technique of using rectangles to approximate the area under a curve is foundational in calculus and leads to the concept of integration, which provides a precise method for calculating such areas.

Use 3 rectangles to approximate the area under $f\left(x\right)=x^2$ from $x=2$to $x=5$ with right endpoints.

• A. 50
• B. 40
• C. 30
• D. 35

Answer: (A)

Use 6 rectangles to approximate the area under $f\left(x\right)=x^3$ from $x=0$ to $x=3$ with left endpoints.

• A. $\frac{225}{16}$
• B. $\frac{225}{8}$
• C. $\frac{81}{4}$
• D. 81

Answer: (A)

Approximating Area Algebraically

In calculus, approximating the area under a curve can be achieved using rectangles, a method known as Riemann sums. When given a function, such as \( f(x) = x^2 \), and an interval, for example, from \( x = 1 \) to \( x = 5 \), we can estimate the area using a specified number of rectangles. In this case, we will use 4 rectangles and the right endpoint method for our approximation.

The first step is to determine the width of each rectangle, denoted as \( \Delta x \). This is calculated using the formula:

\[ \Delta x = \frac{b - a}{n} \]

where \( b \) is the end of the interval (5), \( a \) is the start of the interval (1), and \( n \) is the number of rectangles (4). Plugging in these values gives:

\[ \Delta x = \frac{5 - 1}{4} = 1 \]

Next, we need to find the heights of the rectangles. For right endpoints, the x-values are calculated as follows:

\[ x_k = a + k \cdot \Delta x \]

for \( k = 1, 2, 3, 4 \). Thus, we find:

• For \( k = 1 \): \( x_1 = 1 + 1 \cdot 1 = 2 \)
• For \( k = 2 \): \( x_2 = 1 + 2 \cdot 1 = 3 \)
• For \( k = 3 \): \( x_3 = 1 + 3 \cdot 1 = 4 \)
• For \( k = 4 \): \( x_4 = 1 + 4 \cdot 1 = 5 \)

Now, we evaluate the function at these x-values to find the heights:

• Height at \( x_1 = 2 \): \( f(2) = 2^2 = 4 \)
• Height at \( x_2 = 3 \): \( f(3) = 3^2 = 9 \)
• Height at \( x_3 = 4 \): \( f(4) = 4^2 = 16 \)
• Height at \( x_4 = 5 \): \( f(5) = 5^2 = 25 \)

With the heights determined, we can now calculate the area of each rectangle. The area \( A \) is given by:

\[ A \approx \sum_{k=1}^{n} f(x_k) \cdot \Delta x \]

Substituting the values we found:

\[ A \approx (4 + 9 + 16 + 25) \cdot 1 = 54 \]

Thus, the approximate area under the curve \( f(x) = x^2 \) from \( x = 1 \) to \( x = 5 \) using 4 rectangles and the right endpoint method is 54. This method illustrates how to use algebraic expressions to approximate areas without relying on graphical representations, reinforcing the concept of Riemann sums in calculus.

Approximating Area Algebraically Example 1

To approximate the area under the curve of the function \( f(x) = 4x^2 \) from \( x = 0 \) to \( x = 8 \) using left endpoints and four rectangles, we start by determining the width of each rectangle. The formula for the width (base) of each rectangle is given by:

\[\Delta x = \frac{b - a}{n}\]

Here, \( b \) is the end of the interval (8), \( a \) is the start of the interval (0), and \( n \) is the number of rectangles (4). Plugging in these values, we find:

\[\Delta x = \frac{8 - 0}{4} = 2\]

Next, we identify the left endpoints for each rectangle. The left endpoint for each rectangle can be calculated as follows:

\[x_1 = a + 0 \cdot \Delta x = 0 + 0 \cdot 2 = 0\]

\[x_2 = a + 1 \cdot \Delta x = 0 + 1 \cdot 2 = 2\]

\[x_3 = a + 2 \cdot \Delta x = 0 + 2 \cdot 2 = 4\]

\[x_4 = a + 3 \cdot \Delta x = 0 + 3 \cdot 2 = 6\]

Now, we can set up the sum to approximate the area under the curve using the formula:

\[\text{Area} \approx \sum_{i=1}^{n} f(x_i) \cdot \Delta x\]

Substituting the values we calculated, we have:

\[\text{Area} \approx 2 \cdot (f(0) + f(2) + f(4) + f(6))\]

Next, we evaluate the function at each left endpoint:

\[f(0) = 4(0)^2 = 0\]

\[f(2) = 4(2)^2 = 4 \cdot 4 = 16\]

\[f(4) = 4(4)^2 = 4 \cdot 16 = 64\]

\[f(6) = 4(6)^2 = 4 \cdot 36 = 144\]

Now, substituting these values back into the area approximation gives:

\[\text{Area} \approx 2 \cdot (0 + 16 + 64 + 144) = 2 \cdot 224 = 448\]

Thus, the approximate area under the curve using left endpoints is \( 448 \) square units. This method effectively demonstrates how to use rectangles to estimate the area under a curve when a graph is not available.

Using Limits to Find Area Under the Curve

To find the exact area underneath the curve of a function, we can utilize the concept of limits in conjunction with the method of Riemann sums, which involves approximating the area using rectangles. Initially, we approximate the area by filling it with rectangles and calculating the area of each rectangle using the formula: Area = base × height. As we increase the number of rectangles, the width of each rectangle decreases, leading to a more accurate approximation of the area under the curve.

To achieve an exact area, we consider the limit as the number of rectangles approaches infinity. This process allows us to refine our approximation to the point where it becomes the exact area. The formula for the width of each rectangle, denoted as Δx, is calculated as:

Δx = \(\frac{b - a}{n}\)

where b is the upper limit of the interval, a is the lower limit, and n is the number of rectangles. For example, if we want to find the area under the curve of the function f(x) = \frac{1}{2}x from x = 0 to x = 6, we first calculate:

Δx = \(\frac{6 - 0}{n} = \frac{6}{n}\)

Next, we determine the height of each rectangle using the function evaluated at specific points. For right endpoints, we find:

x_k = a + kΔx = 0 + k\(\frac{6}{n}\) = \(\frac{6k}{n}\)

Thus, the height of each rectangle is:

f(x_k) = f\(\left(\frac{6k}{n}\right)\) = \(\frac{1}{2}\left(\frac{6k}{n}\right) = \frac{3k}{n}\)

Now, we can express the area as a summation:

Area = \(\lim_{n \to \infty} \sum_{k=1}^{n} f(x_k)Δx\)

Substituting the values we calculated, we have:

Area = \(\lim_{n \to \infty} \sum_{k=1}^{n} \left(\frac{3k}{n}\right)\left(\frac{6}{n}\right)\)

After simplifying, this becomes:

Area = \(\lim_{n \to \infty} \frac{18}{n^2} \sum_{k=1}^{n} k\)

Using the formula for the sum of the first n integers, \(\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}\), we substitute this into our equation:

Area = \(\lim_{n \to \infty} \frac{18}{n^2} \cdot \frac{n(n + 1)}{2}\)

After simplifying further, we find:

Area = \(\lim_{n \to \infty} \frac{9(n + 1)}{n}\)

As n approaches infinity, the limit evaluates to:

Area = 9

This result indicates that the exact area underneath the curve of the function f(x) = \frac{1}{2}x from x = 0 to x = 6 is 9 square units. This method demonstrates how the limit definition of integrals provides a powerful tool for calculating exact areas under curves.

Find the area under the curve, of the function $f\left(x\right)=3x+2$ from $x=0$ to $x=4$ using limits.

• A. 32
• B. 56
• C. 26
• D. 50

Answer: (A)

Find the area under the curve, of the function $f\left(x\right)=x^2$ from $x=0$ to $x=8$ using limits.

• A. 256
• B. $512$
• C. $\frac{1}{3}$
• D. $\frac{512}{3}$

Answer: (D)

Here’s what students ask on this topic:

How do you find the area under a linear function using geometric shapes?

To find the area under a linear function, you can use familiar geometric shapes like triangles and rectangles. For example, if you have a linear function such as f(x) = x, and you need to find the area from x = 0 to x = 5, you can visualize the area as a triangle. The base of the triangle is the interval on the x-axis, and the height is the value of the function at the endpoint. The area of a triangle is calculated using the formula: $\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$. In this case, the base and height are both 5, so the area is $\frac{1}{2}\times 5\times 5=12.5$.

What is the method for approximating the area under a curve using rectangles?

To approximate the area under a curve, especially for non-linear functions, you can use the method of rectangles. This involves dividing the area into a series of rectangles, calculating the area of each, and summing them up. The base of each rectangle is determined by dividing the interval into equal parts, and the height is determined by the function value at either the left or right endpoint of each subinterval. The area of each rectangle is calculated using the formula: $\mathrm{base}\times \mathrm{height}$. By increasing the number of rectangles, the approximation becomes more accurate. This method is known as the Riemann sum, and it can be used with left or right endpoints, or even midpoints, to improve accuracy.

How can you find the exact area under a curve using the limit definition?

To find the exact area under a curve, you can use the limit definition of the integral. This involves taking the limit of the Riemann sum as the number of rectangles approaches infinity. The formula used is: $\sum _{n\to \infty }\frac{f}{\left(}$. Here, $\Delta x$ is the width of each rectangle, and $f\left(x\right)$ is the function value at a specific point. By applying the limit, the sum of the areas of the rectangles converges to the exact area under the curve, which is the definite integral of the function over the given interval.

What is the difference between left and right endpoint approximations?

Left and right endpoint approximations are methods used to estimate the area under a curve using rectangles. In a left endpoint approximation, the height of each rectangle is determined by the function value at the left endpoint of each subinterval. Conversely, in a right endpoint approximation, the height is determined by the function value at the right endpoint. The choice between left and right endpoints affects the accuracy of the approximation. Left endpoint approximations tend to underestimate the area if the function is increasing, while right endpoint approximations tend to overestimate it. By using more rectangles, both methods can provide a closer approximation to the true area under the curve.

How do you calculate the base of rectangles when approximating the area under a curve?

When approximating the area under a curve using rectangles, the base of each rectangle is calculated by dividing the interval of interest into equal parts. The formula for the base is: $\frac{b-a}{n}$, where $a$ and $b$ are the endpoints of the interval, and $n$ is the number of rectangles. This ensures that each rectangle has the same width, allowing for a consistent approximation. As the number of rectangles increases, the base becomes smaller, leading to a more accurate approximation of the area under the curve.