Understanding the Trapezoidal Rule for Approximating Definite Integrals

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In summary, the trapezoidal rule is a method for approximating definite integrals by representing the area under a curve as a series of trapezoids. The equation for this rule is h[(1/2)y0 + y1 + y2 + ... + yn-1 + (1/2)yn], where h is the width of each trapezoid and the 1/2 is multiplied to the first and last y values. This is because the 1/2 represents the area of the two triangles at the ends of the series, which are not accounted for in the rest of the equation.
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MMM
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Homework Statement


I'm curious about how the trapezoidal rule is derived for approximating definite integrals.

Homework Equations



According to my calculus book the equation is $$h[(1/2)y_{0} + y_{1} + y_{2} + ... +y_{n-1} + (1/2)y_{n}]$$

The Attempt at a Solution


I'm curious as to why the first and last y values are multiplied by $$1/2$$
I've solved a lot of problems using the trapezoidal rule, but I don't quite understand it. Any insight on why the first and last y values are multiplied by $$1/2$$.
 
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MMM said:

Homework Statement


I'm curious about how the trapezoidal rule is derived for approximating definite integrals.

Homework Equations



According to my calculus book the equation is $$h[(1/2)y_{0} + y_{1} + y_{2} + ... +y_{n-1} + (1/2)y_{n}]$$

The Attempt at a Solution


I'm curious as to why the first and last y values are multiplied by $$1/2$$
I've solved a lot of problems using the trapezoidal rule, but I don't quite understand it. Any insight on why the first and last y values are multiplied by $$1/2$$.
You are representing the area under the curve as a set of trapezoids. The total area is h(y0 + y1)/2 + h(y1 + y2)/2 ... and so on. The 1/2 goes away for all points but the first and last.
 
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I get it now, I appreciate the help.
 

Related to Understanding the Trapezoidal Rule for Approximating Definite Integrals

1. What is intuition and why is it important in science?

Intuition is a gut feeling or instinctual knowledge that is not based on conscious reasoning. In science, it is important because it allows scientists to make connections and come up with new ideas and hypotheses that may not be obvious through traditional methods of scientific inquiry.

2. Can intuition be trusted in scientific research?

Intuition is just one tool in a scientist's repertoire and should not be solely relied upon. It should be used in conjunction with other methods, such as experimentation and analysis, to validate its validity.

3. How can I develop my intuition as a scientist?

Intuition is often a result of years of experience and knowledge in a particular field. To develop your intuition, it is important to constantly learn and stay current with research and findings in your area of expertise. It is also helpful to engage in practices such as meditation and journaling to tap into your subconscious and improve your ability to make intuitive connections.

4. Are some scientists more intuitive than others?

Some scientists may have a natural inclination towards intuition, but it is a skill that can be developed and improved upon through practice and experience. It is also important to note that intuition should not be favored over evidence-based methods in scientific research.

5. How does intuition play a role in the scientific method?

Intuition can play a role in the initial stages of the scientific method, such as in forming a hypothesis or generating new ideas. However, it should not be used as the sole basis for conclusions or findings. The scientific method relies on empirical evidence and logical reasoning to test and validate hypotheses.

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