Area Under Sine Curve: A Step-by-Step Calculation

by Mei Lin 50 views

Hey guys! Ever wondered how to calculate the area under a curve? It might sound intimidating, but it's actually a pretty cool concept with lots of real-world applications. In this article, we're going to dive deep into calculating the area under the sine curve, specifically from 0 to 90 degrees (or 0 to PI/2 radians). We'll break it down step-by-step, so you can understand the process and even apply it to other curves. So, grab your calculators and let's get started!

Understanding the Problem: Summing Areas of Rectangles

Before we jump into the math, let's make sure we understand the problem. We're trying to find the total area trapped between the sine curve (sin(x)), the x-axis, and the vertical lines at x = 0 and x = PI/2. Now, there are fancy calculus methods to do this, but we're going to take a more visual approach: we'll approximate the area by dividing it into a bunch of rectangles and adding up their areas.

Imagine slicing the area under the curve into nine equal vertical strips. Each strip will form the base of a rectangle. The height of each rectangle will be determined by the value of the sine function at some point within that strip. By calculating the area of each rectangle (base times height) and then summing them up, we get an approximation of the total area under the curve. The more rectangles we use, the better our approximation becomes. Think of it like building a mosaic – the smaller the tiles (rectangles), the smoother the image (the area under the curve).

This method, called the Riemann sum, is a fundamental concept in calculus and provides a powerful way to understand integration. We're essentially approximating the definite integral of the sine function using a finite sum. It's a hands-on way to see how the concept of integration works, and it's a great stepping stone to understanding more advanced techniques.

Key Takeaways:

  • We're finding the area under the sine curve from 0 to PI/2.
  • We're approximating this area by summing the areas of nine rectangles.
  • This method is called the Riemann sum, a core concept in calculus.

Breaking Down the Calculation: Step-by-Step

Okay, let's get down to the nitty-gritty. To calculate the sum of the areas of these nine rectangles, we'll follow these steps:

  1. Divide the interval: We need to divide the interval from 0 to PI/2 into nine equal subintervals. The width of each subinterval (which will be the base of our rectangles) is calculated by dividing the total interval length (PI/2 - 0 = PI/2) by the number of rectangles (9). So, the width of each rectangle (Δx) is (PI/2) / 9 = PI/18.

  2. Choose a sample point: Within each subinterval, we need to choose a point where we'll evaluate the sine function to determine the height of the rectangle. There are different ways to choose this point (left endpoint, right endpoint, midpoint), but for this example, let's use the right endpoint. This means the x-coordinate for the height of the first rectangle will be PI/18, for the second rectangle 2PI/18, and so on.

  3. Calculate the height: For each rectangle, we'll plug the x-coordinate of our chosen point into the sine function to find the height. For example, the height of the first rectangle will be sin(PI/18), the height of the second rectangle will be sin(2PI/18), and so on.

  4. Calculate the area of each rectangle: The area of each rectangle is simply its base (Δx = PI/18) multiplied by its height (sin(x) at the chosen point).

  5. Sum the areas: Finally, we add up the areas of all nine rectangles to get our approximation of the total area under the curve. This sum is the Riemann sum we're looking for.

Let's put this into a more mathematical form. If we let xᵢ represent the right endpoint of the i-th subinterval, then the area of the i-th rectangle is given by:

Areaᵢ = Δx * sin(xᵢ) = (PI/18) * sin(i * PI/18)

The total approximate area (A) is then the sum of these individual areas:

A ≈ Σᵢ₌₁⁹ Areaᵢ = Σᵢ₌₁⁹ [(PI/18) * sin(i * PI/18)]

This formula might look a bit intimidating, but it's just a concise way of expressing the process we've outlined. We're essentially calculating the sine of nine different angles (PI/18, 2PI/18, ..., 9PI/18), multiplying each by PI/18, and then adding the results together.

Key Takeaways:

  • We divided the interval into nine equal subintervals, each with a width of PI/18.
  • We chose the right endpoint of each subinterval to determine the height of the rectangle.
  • We used the formula A ≈ Σᵢ₌₁⁹ [(PI/18) * sin(i * PI/18)] to calculate the approximate area.

Putting It All Together: The Calculation and Result

Now for the fun part – the actual calculation! We need to evaluate the sum:

A ≈ (PI/18) * [sin(PI/18) + sin(2PI/18) + sin(3PI/18) + sin(4PI/18) + sin(5PI/18) + sin(6PI/18) + sin(7PI/18) + sin(8PI/18) + sin(9PI/18)]

This is where a calculator comes in handy (especially one that can handle trigonometric functions in radians!). Let's break it down:

  1. Calculate the sine of each angle:

    • sin(PI/18) ≈ 0.1736
    • sin(2PI/18) ≈ 0.3420
    • sin(3PI/18) ≈ 0.5000
    • sin(4PI/18) ≈ 0.6428
    • sin(5PI/18) ≈ 0.7660
    • sin(6PI/18) ≈ 0.8660
    • sin(7PI/18) ≈ 0.9397
    • sin(8PI/18) ≈ 0.9848
    • sin(9PI/18) = sin(PI/2) = 1.0000

  2. Sum the sine values:

      1. 1736 + 0.3420 + 0.5000 + 0.6428 + 0.7660 + 0.8660 + 0.9397 + 0.9848 + 1.0000 ≈ 6.2149

  3. Multiply by PI/18:

    • (PI/18) * 6.2149 ≈ (3.1416/18) * 6.2149 ≈ 1.084

So, our approximation for the area under the sine curve from 0 to PI/2, using nine rectangles and the right endpoint rule, is approximately 1.084 square units.

Key Takeaways:

  • We calculated the sine of each angle and summed the values.
  • We multiplied the sum by PI/18 to get our approximate area.
  • Our approximation is about 1.084 square units.

How Accurate Is Our Approximation?

Now, you might be wondering, how close is this approximation to the actual area under the curve? Well, remember that we're using rectangles to approximate a curved shape, so there's going to be some error. In our case, because we used the right endpoint rule, we're slightly overestimating the area. This is because the rectangles extend slightly above the sine curve in each subinterval.

The actual area under the sine curve from 0 to PI/2 can be found using calculus (specifically, definite integration). The definite integral of sin(x) from 0 to PI/2 is equal to 1. So, our approximation of 1.084 is about 8.4% higher than the true value.

To get a more accurate approximation, we could do a few things:

  • Use more rectangles: The more rectangles we use, the smaller the width of each rectangle, and the better the approximation will be. As the number of rectangles approaches infinity, the Riemann sum approaches the definite integral, and our approximation becomes exact.
  • Use a different sampling method: Instead of the right endpoint rule, we could use the left endpoint rule or the midpoint rule. The midpoint rule often gives a more accurate approximation because it tends to balance out overestimates and underestimates within each subinterval.

Key Takeaways:

  • Our approximation is slightly higher than the actual area due to the right endpoint rule.
  • We can improve the accuracy by using more rectangles or a different sampling method.
  • The actual area under the sine curve from 0 to PI/2 is 1.

Applications and Further Exploration

Calculating the area under a curve isn't just a mathematical exercise; it has numerous applications in various fields:

  • Physics: Finding the distance traveled by an object given its velocity function.
  • Engineering: Calculating the work done by a force.
  • Economics: Determining consumer surplus or producer surplus.
  • Probability: Finding the probability of an event within a continuous distribution.

The concept of Riemann sums and integration is a cornerstone of calculus and has far-reaching implications. By understanding how to approximate the area under a curve using rectangles, you've taken a significant step towards mastering these powerful mathematical tools.

Further Exploration:

  • Try calculating the area under the sine curve using different numbers of rectangles (e.g., 18, 36, 100) and see how the approximation improves.
  • Experiment with the left endpoint rule and the midpoint rule and compare the results.
  • Explore other curves and try to approximate the area under them using Riemann sums.
  • Dive into the world of definite integrals and learn how to calculate areas exactly using calculus.

Calculating the area under the sine curve using Riemann sums is a fantastic way to visualize and understand the fundamental concepts of calculus. It's a hands-on approach that bridges the gap between abstract theory and concrete applications. So, keep practicing, keep exploring, and keep those mathematical muscles flexing!