Problem 1 - Calculus

Question

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{2}} d x $$

Step-by-Step Solution

Answer

The exact integral value is \( \frac{2}{3} \), and numerical approximations should be compared against this.

Step 1: Define the function and interval

The function given is \( f(x) = \frac{1}{x^2} \) and the interval is from \( x = 1 \) to \( x = 3 \). We'll use different numerical methods to approximate the integral over this interval.

Step 2: Apply Left Riemann Sum

To approximate the integral using the Left Riemann Sum with \( n = 8 \), we take the left end of each subinterval to calculate the heights of the rectangles. The width \( \Delta x = \frac{3-1}{8} = 0.25 \). Sum up all values: \( \sum_{i=0}^{7} f(x_i) \times 0.25 \) where \( x_i = 1 + 0.25i \). Evaluate each \( f(x_i) \) and compute the sum.

Step 3: Apply Right Riemann Sum

For the Right Riemann Sum, use the right endpoint of each subinterval. Again, \( \Delta x = 0.25 \) leading to sum \( \sum_{i=1}^{8} f(x_i) \times 0.25 \) where \( x_i = 1 + 0.25i \). Calculate each \( f(x_i) \) for \( i \) from 1 to 8 and sum them.

Step 4: Apply Trapezoidal Rule

The Trapezoidal Rule formula is \( T = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \). Here, use the same \( x_i \) values and sum them according to the Trapezoidal Rule formula.

Step 5: Apply Parabolic Rule (Simpson's Rule)

For Simpson's Rule with \( n = 8 \), the formula is \( S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{odd\ i} f(x_i) + 2 \sum_{even\ i} f(x_i) + f(x_n) \right) \). Use the function values at appropriate \( x_i \) points to compute \( S \).

Step 6: Apply Second Fundamental Theorem of Calculus

To find the exact integral value, evaluate \( \int_{1}^{3} \frac{1}{x^2} \, dx \) by computing the antiderivative \( F(x) = -\frac{1}{x} \), and then \( F(3) - F(1) = -\frac{1}{3} + 1 = \frac{2}{3} \).

Step 7: Compare Results

Compare the approximate values with the exact value \( \frac{2}{3} \) to see which numerical method gives a result closest to the exact solution.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that allows us to find the area under a curve between two points. It's represented as \( \int_{a}^{b} f(x) \, dx \) where \( a \) and \( b \) are the bounds of integration, and \( f(x) \) is the function you're integrating. The result of a definite integral gives the net area between the curve and the x-axis from \( x = a \) to \( x = b \). This net area helps us measure quantities like displacement, total accumulation, or the overall value of a function over a defined interval. To understand this better, think of the curve as a physical surface, and the definite integral as a way to calculate its precise area. For functions that aren't easy to integrate directly, numerical methods can approximate these integrals. These methods, such as Riemann Sum, Trapezoidal Rule, and Simpson's Rule, break the curve into simpler shapes to estimate the area.

Riemann Sum

The Riemann Sum is one of the most fundamental methods used for approximating the definite integral of a function. It involves partitioning the domain into subintervals, then summing up the areas of rectangles whose heights correspond to the function value at specific points within these intervals. You can choose different evaluation points for the height of these rectangles, namely the left endpoint, right endpoint, or even a midpoint, giving rise to different types of Riemann sums.

• Left Riemann Sum: Uses the left endpoint of each subinterval to determine the height of the rectangle.
• Right Riemann Sum: Uses the right endpoint of each subinterval, potentially giving different (often more accurate) results.

The choice between these depends on your preference or required accuracy. For both methods, as the number of subintervals increases, the approximation typically improves, becoming closer to the true value of the definite integral.

Trapezoidal Rule

The Trapezoidal Rule is a step up from the Riemann Sum and offers a more refined approximation. Instead of rectangles, it approximates the area under the curve using trapezoids, which more closely follow the contour of the curve. Each trapezoid is formed by using both endpoints of each subinterval as its corners.The formula for the Trapezoidal Rule is:\[T = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)\]This method generally provides a better approximation than using rectangles, especially if the function is smooth. It takes into account the slope of the curve between the endpoints, often leading to quicker convergence to the actual value for integrals. This makes the Trapezoidal Rule especially useful for functions that change rapidly.

Simpson's Rule

Simpson's Rule further refines numerical integration by using parabolic segments instead of straight lines or rectangles. This method is highly accurate and works best when the function is smooth, capturing its curve more naturally.For Simpson's Rule, the formula is:\[S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{odd\ i} f(x_i) + 2 \sum_{even\ i} f(x_i) + f(x_n) \right)\]This formula considers both the odd and even subinterval values, giving greater weight to them by multiplying by 4 and 2, respectively. Simpson's Rule is often chosen when high precision is needed with fewer subintervals, as it quickly converges to the exact value of the definite integral.

Fundamental Theorem of Calculus

At the heart of calculus, the Fundamental Theorem links differentiation and integration, providing a new perspective on finding exact values of definite integrals. It states that if \( F(x) \) is the antiderivative of \( f(x) \), then:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]This theorem allows us to compute an integral exactly by identifying the antiderivative of the function and evaluating it at the upper and lower limits of integration. In the given exercise, by finding the antiderivative \( F(x) = -\frac{1}{x} \) and evaluating at the bounds, we determined the exact area under the curve.Understanding this concept is crucial because it offers a deeper insight into both the mechanics and the beautifully ordered world of calculus. It underscores how derivatives and integrals are inverse processes, bringing together the entire discipline under a unified framework.