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Step 1:I'll explain how to find the trapezoidal sum with a clear, step-by-step solution:
Step 2:: Understand the Trapezoidal Sum
The trapezoidal sum is a numerical integration method used to approximate the definite integral of a function by dividing the area under a curve into trapezoids.
Step 3:: Define the Trapezoidal Sum Formula
- $$f(x)$$ is the function being integrated
The general formula for the trapezoidal sum is: Where:
Step 4:: Calculation Process
Step 5:
Divide the interval $$[a,b]$$ into $$n$$ equal subintervals
Step 6:
Calculate the width of each subinterval: $$h = \frac{b-a}{n}
Step 7:
Compute the x-coordinates: $$x_i = a + ih$$ for $$i = 0, 1, 2, \ldots, n
Step 8:
Evaluate the function at each x-coordinate
Step 9:
Apply the trapezoidal sum formula
Step 10:: Example Calculation
T_4 = \frac{0.25}{2}\left[0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1\right]
- a = 0, b = 1, n = 4 Calculating function values: Applying the formula:
Final Answer
The trapezoidal sum provides an approximation of the definite integral by averaging the areas of trapezoids formed under the curve. More subintervals (larger n) typically result in a more accurate approximation. Key Insights: - The method works by replacing the curve with a series of trapezoids - Increasing the number of subintervals improves accuracy - It's particularly useful when the exact integral is difficult to compute analytically
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