91Ó°ÊÓ

Definite Integrals are a fundamental concept in calculus used to find the area under a curve or the accumulation of quantities over a given interval. The definite integral of a function \(f(x)\) from \(a\) to \(b\), represented as \(\int_{a}^{b} f(x) \, dx\), calculates the net "signed" area between the function and the x-axis over that interval.

Some key elements of definite integrals include:

• Antiderivative: The process of finding a function whose derivative equals the original function.
• Limits of Integration: The values of \(a\) and \(b\) that define the interval over which we calculate the area.
• Evaluation: After finding the antiderivative \(F(x)\), evaluate it at the endpoints \(a\) and \(b\) and subtract: \(F(b) - F(a)\).

In the exercise, the definite integral \(\int_{0}^{1} 7x^6 \, dx\) was evaluated to find an exact integral value of 1. This shows how definite integrals can precisely determine the area under a curve.

Riemann Sums are an essential tool for estimating the area under a curve when computing definite integrals. They serve as a bridge between discrete summation and continuous integration. By partitioning an interval into subintervals and using the heights of rectangles to approximate the area under the curve, Riemann Sums provide a practical way to approximate integrals.

There are different types of Riemann sums based on which point in the subinterval is used to calculate the height of the rectangles:

• Left Riemann Sum: Uses the left endpoint of each subinterval.
• Right Riemann Sum: Uses the right endpoint of each subinterval.
• Midpoint Riemann Sum: Uses the midpoint of each subinterval.

In the given exercise, LEFT(5), RIGHT(5), and MID(5) were calculated to approximate \(\int_{0}^{1} 7x^6 \, dx\). Increasing the number of subintervals (from \(n=5\) to \(n=10\)) generally provides better approximations.

The Trapezoidal Rule is another numerical integration method. It approximates the region under a curve by dividing it into trapezoids rather than rectangles. This can often lead to more accurate approximations than Riemann Sums, particularly when the graph of the function is relatively smooth.

To apply the Trapezoidal Rule, consider:

• Divide the interval from \(a\) to \(b\) into \(n\) subintervals.
• Each trapezoid uses the left and right endpoints of subintervals as its bases.
• The formula is \(\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\), where \(\Delta x\) is the width of each subinterval.

In the exercise, TRAP(5) was calculated as an average of LEFT(5) and RIGHT(5), illustrating its relation to Riemann sums and providing a better estimate of the integral.

Simpson's Rule is a powerful method for numerical integration that uses parabolic arcs instead of straight lines to approximate the curve of a function. This makes it particularly useful for smooth functions, yielding very accurate approximations.

Simpson's Rule is applied as follows:

• The interval \([a, b]\) is divided into an even number of subintervals \(n\).
• The rule is given by \(\frac{\Delta x}{3}(f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n))\).
• It considers both midpoints and endpoints, weighting them differently for precision.

In the exercise, SIMP(10) was used to find the approximation of \(\int_{0}^{1} 7x^6 \, dx\), highlighting its accuracy over methods like the Trapezoidal Rule when using a greater number of subintervals.

Error Analysis in numerical integration helps us understand the accuracy of different approximation methods by evaluating how close the estimates are to the exact integral. Calculating errors allows us to choose the best method for a given function and desired accuracy.

To perform error analysis:

• Compute the absolute error as \(|\text{Approximation} - \text{Exact value}|\).
• Compare errors across different methods (LEFT, RIGHT, MID, TRAP, SIMP) and different subintervals (\(n=5\) vs. \(n=10\)).
• Calculate the error ratio \(\frac{\text{Error for } n=5}{\text{Error for } n=10}\) which often demonstrates reduced error with increased subintervals.

In the exercise, the error ratios for each method confirmed that increasing \(n\) generally reduces errors, aligning with theoretical expectations in numerical analysis.