Understanding the Trapezoidal Rule for Approximating Definite Integrals

[MMM]

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 $$1/2$$.

 

[Doc Al]

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.

 

[MMM]

I get it now, I appreciate the help.