Riemann Sum: Integral Calculation With N=3

Hey guys! Today, let's dive into the world of Riemann sums, a fundamental concept in calculus that allows us to approximate definite integrals. We'll be working through a specific example, breaking it down step-by-step, so you can really get a handle on how it all works. We are going to focus on how to approximate the definite integral of a function using Riemann sums, specifically with right and left endpoints.

Problem Statement

We are given the definite integral:

$$\int_1^4 (2x^2 + 4x + 5) dx$$

Our mission is to approximate this integral using Riemann sums with $n = 3$ subintervals, first using right endpoints and then left endpoints.

(a) Riemann Sum with Right Endpoints ($R_3$)

So, what's a Riemann sum? Simply put, it's an approximation of the area under a curve using rectangles. The more rectangles we use (the larger n is), the better our approximation becomes. When we use right endpoints, the height of each rectangle is determined by the function's value at the right edge of each subinterval.

Step 1: Determine the width of each subinterval ($\Delta x$)

First, we need to figure out how wide each of our three rectangles will be. We can calculate the width ($\Delta x$) using the formula:

$$\Delta x = \frac{b - a}{n}$$

Where:

• $a$ is the lower limit of integration (in our case, 1).
• $b$ is the upper limit of integration (in our case, 4).
• $n$ is the number of subintervals (rectangles) we're using (in our case, 3).

Plugging in our values, we get:

$$\Delta x = \frac{4 - 1}{3} = \frac{3}{3} = 1$$

So, each rectangle will have a width of 1.

Step 2: Determine the right endpoints of each subinterval

Since our interval is $[1, 4]$ and $\Delta x = 1$, our subintervals are:

• $[1, 2]$
• $[2, 3]$
• $[3, 4]$

The right endpoints of these subintervals are 2, 3, and 4. These are the x-values we'll use to determine the height of our rectangles.

Step 3: Evaluate the function at each right endpoint

Our function is $f(x) = 2x^2 + 4x + 5$. We need to find the value of the function at each of our right endpoints:

• $f(2) = 2(2)^2 + 4(2) + 5 = 8 + 8 + 5 = 21$
• $f(3) = 2(3)^2 + 4(3) + 5 = 18 + 12 + 5 = 35$
• $f(4) = 2(4)^2 + 4(4) + 5 = 32 + 16 + 5 = 53$

Step 4: Calculate the Riemann sum

The Riemann sum using right endpoints ($R_3$) is calculated as follows:

$$R_3 = \Delta x [f(2) + f(3) + f(4)]$$

Plugging in the values we calculated:

$$R_3 = 1 [21 + 35 + 53] = 1 [109] = 109$$

Therefore, the Riemann sum for this integral using right endpoints and $n = 3$ is $R_3 = 109$.

(b) Riemann Sum with Left Endpoints ($L_3$)

Now, let's calculate the Riemann sum using left endpoints. The process is very similar, but this time, the height of each rectangle is determined by the function's value at the left edge of each subinterval.

Step 1: Determine the width of each subinterval ($\Delta x$)

As before, $\Delta x = 1$ since the interval and $n$ are unchanged.

Step 2: Determine the left endpoints of each subinterval

Our subintervals are still:

• $[1, 2]$
• $[2, 3]$
• $[3, 4]$

But this time, we're interested in the left endpoints, which are 1, 2, and 3.

Step 3: Evaluate the function at each left endpoint

We need to find the value of our function, $f(x) = 2x^2 + 4x + 5$, at each of these left endpoints:

• $f(1) = 2(1)^2 + 4(1) + 5 = 2 + 4 + 5 = 11$
• $f(2) = 2(2)^2 + 4(2) + 5 = 8 + 8 + 5 = 21$
• $f(3) = 2(3)^2 + 4(3) + 5 = 18 + 12 + 5 = 35$

Step 4: Calculate the Riemann sum

The Riemann sum using left endpoints ($L_3$) is calculated as follows:

$$L_3 = \Delta x [f(1) + f(2) + f(3)]$$

Plugging in the values we calculated:

$$L_3 = 1 [11 + 21 + 35] = 1 [67] = 67$$

Therefore, the Riemann sum for this integral using left endpoints and $n = 3$ is $L_3 = 67$.

Summary of Results

• Riemann sum with right endpoints ($R_3$): 109
• Riemann sum with left endpoints ($L_3$): 67

Visualizing the Riemann Sums

Imagine drawing the graph of $f(x) = 2x^2 + 4x + 5$ from $x = 1$ to $x = 4$. The Riemann sum with right endpoints overestimates the area under the curve because the rectangles extend slightly above the curve in each subinterval. Conversely, the Riemann sum with left endpoints underestimates the area because the rectangles fall slightly below the curve. This gives you a basic intuition of how Riemann sums work. By increasing the number of rectangles ($n$), the approximation gets better and better, and it approaches the exact value of the definite integral. The exact value will lie between $L_3$ and $R_3$.

Key Takeaways

• Riemann sums provide a method for approximating definite integrals.
• Right endpoints tend to overestimate the integral, while left endpoints tend to underestimate it.
• Increasing the number of subintervals ($n$) improves the accuracy of the approximation.
• The width of each subinterval ($\Delta x$) is crucial for calculating the Riemann sum.
• Understanding Riemann sums is foundational for grasping the concept of the definite integral as a limit.

Why are Riemann Sums Important?

Riemann sums are not just theoretical exercises! They form the basis of how computers and numerical methods approximate integrals. In many real-world applications, finding an exact solution to an integral is impossible. Think about calculating the area of a lake on a map, or the total energy consumption over a period of time. Riemann sums (or more sophisticated numerical integration techniques derived from them) provide a practical way to get a close estimate. They are also essential for understanding the fundamental theorem of calculus, which connects integration and differentiation. Without grasping the concept of approximating areas with sums, the leap to understanding integration as the reverse of differentiation becomes much harder. So, while it might seem like a lot of rectangles, mastering Riemann sums opens the door to more advanced topics and real-world problem-solving!

So, there you have it! We've successfully calculated the Riemann sums for our given integral using both right and left endpoints. Keep practicing, and you'll become a Riemann sum master in no time! Remember the key is to visualize what you're doing – draw the rectangles and see how they approximate the area under the curve. Good luck, and happy calculating!