Problem 55
Question
Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b] .\) To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=2 x+2 ; a=-1, b=1 $$
Step-by-Step Solution
Verified Answer
The area under the curve is 4.
1Step 1: Define the Function and Interval
We are given the function \( y = 2x + 2 \) and the interval \([-1, 1]\). Our goal is to find the area under this curve over the given interval.
2Step 2: Divide the Interval into Subintervals
Divide the interval \([-1, 1]\) into \(n\) equal subintervals. The width of each subinterval is \( \Delta x = \frac{b-a}{n} = \frac{1 - (-1)}{n} = \frac{2}{n} \).
3Step 3: Estimate the Area with Rectangles
We can approximate the area using rectangles. The height of each rectangle can be estimated using the function value at the right endpoint of each subinterval. The height at the \(i^{th}\) subinterval is \(f\left(a + i \Delta x\right) = 2(a + i \Delta x) + 2\).
4Step 4: Compute the Area of Each Rectangle
The area of each rectangle is the product of its height and width: \[ A_i = f\left(a + i \Delta x\right) \Delta x = \left(2(-1 + i \frac{2}{n}) + 2\right) \cdot \frac{2}{n} \].
5Step 5: Sum the Areas of All Rectangles
The approximate area under the curve is the sum of all rectangles: \[ A_n = \sum_{i=1}^{n} \left(2(-1 + i \frac{2}{n}) + 2\right) \cdot \frac{2}{n} \].
6Step 6: Evaluate the Limit as n Approaches Infinity
To find the exact area, take the limit as \(n\) approaches infinity: \[ \lim_{n \to \infty} \sum_{i=1}^{n} \left(2(-1 + i \frac{2}{n}) + 2\right) \cdot \frac{2}{n} = \int_{-1}^{1} (2x + 2) \, dx \].
7Step 7: Calculate the Definite Integral
Compute the integral: \[ \int_{-1}^{1} (2x + 2) \, dx = \left[ x^2 + 2x \right]_{-1}^{1} = (1^2 + 2 \times 1) - ((-1)^2 + 2 \times (-1)) = 3 - (-1) = 4 \].
Key Concepts
Area Under a CurveRectangle Approximation MethodLimit of a Sum
Area Under a Curve
In mathematics, when we refer to the "area under a curve," we are talking about the integral of a function over a certain interval. This integral provides us with a way to calculate the total accumulation, much like finding the area of a shape.
To find the area under the curve of a function like \(y = f(x)\), over a closed interval \([a, b]\), we use the definite integral \(\int_a^b f(x) \; dx\). This integral gives us the area bound between the graph of the function, the x-axis, and the vertical lines at \(x = a\) and \(x = b\). Simply put, it's the enclosed space under the line that the function plots on the graph.
For instance, in the context of the exercise, we are asked to find the area under the linear curve \(y = 2x + 2\) from \(x = -1\) to \(x = 1\). This is solved by calculating the definite integral over this interval, providing an exact numeral value to the area under the curve.
To find the area under the curve of a function like \(y = f(x)\), over a closed interval \([a, b]\), we use the definite integral \(\int_a^b f(x) \; dx\). This integral gives us the area bound between the graph of the function, the x-axis, and the vertical lines at \(x = a\) and \(x = b\). Simply put, it's the enclosed space under the line that the function plots on the graph.
For instance, in the context of the exercise, we are asked to find the area under the linear curve \(y = 2x + 2\) from \(x = -1\) to \(x = 1\). This is solved by calculating the definite integral over this interval, providing an exact numeral value to the area under the curve.
Rectangle Approximation Method
The rectangle approximation method is a technique to estimate the area under a curve using a series of rectangles. This method becomes essential when a function is too complex to integrate directly or when calculating a specific point for approximation or educational purposes.
This method involves dividing the region under a curve into n equal subintervals and drawing rectangles, having their heights determined by the function's value at selected points within these subintervals. In our exercise, these rectangles' heights are defined at the right endpoints of the subintervals, yielding a straightforward way to measure the space slightly above the curve. Each rectangle stands on its width \(\Delta x = \frac{b-a}{n}\).
By summing up the areas of these rectangles, we approximate the curve's area. The more rectangles used (higher \(n\) values), the more accurate this approximation becomes, eventually leading us to the precise area as \(n \rightarrow \infty\).
This method involves dividing the region under a curve into n equal subintervals and drawing rectangles, having their heights determined by the function's value at selected points within these subintervals. In our exercise, these rectangles' heights are defined at the right endpoints of the subintervals, yielding a straightforward way to measure the space slightly above the curve. Each rectangle stands on its width \(\Delta x = \frac{b-a}{n}\).
By summing up the areas of these rectangles, we approximate the curve's area. The more rectangles used (higher \(n\) values), the more accurate this approximation becomes, eventually leading us to the precise area as \(n \rightarrow \infty\).
Limit of a Sum
Adding up a large number of small things is often done by taking the limit of a sum. In calculus, we use this idea to transition from an approximation into precision, particularly when estimating areas, volumes, or other quantities defined by functions.
This concept is critical when using methods like the rectangle approximation method, which calculates the approximate area under a curve by adding up the areas of all rectangles. As the number of rectangles \(n\) increases, the width \(\Delta x\) of each rectangle decreases, and the approximation becomes closer to the actual area.
The process involves considering the limit \(\lim_{n \to \infty} \sum_{i=1}^{n} A_i\) where \(A_i\) represents the area of each rectangle. As \(n\) grows larger, the sum of the areas of these rectangles approaches the value of the definite integral \(\int_a^b f(x) \; dx\), effectively "smoothing out" the straight lines to follow the curve's precise path. This approach is beautifully encapsulated in the fundamental theorem of calculus, bridging connections between derivatives, integrals, and limits.
This concept is critical when using methods like the rectangle approximation method, which calculates the approximate area under a curve by adding up the areas of all rectangles. As the number of rectangles \(n\) increases, the width \(\Delta x\) of each rectangle decreases, and the approximation becomes closer to the actual area.
The process involves considering the limit \(\lim_{n \to \infty} \sum_{i=1}^{n} A_i\) where \(A_i\) represents the area of each rectangle. As \(n\) grows larger, the sum of the areas of these rectangles approaches the value of the definite integral \(\int_a^b f(x) \; dx\), effectively "smoothing out" the straight lines to follow the curve's precise path. This approach is beautifully encapsulated in the fundamental theorem of calculus, bridging connections between derivatives, integrals, and limits.
Other exercises in this chapter
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