(a) Verify that the average of the left and right end-point...

(a) Verify that the average of the left and right end-point approximations as given in Table 7.7 .1 gives Formula ( 2 ) for the trapezoidal approximation. (b) Suppose that $f$ is a continuous nonnegative function on the interval $(a, b]$ and partition $[a, b]$ with equally spaced points, $a=x_{0}<x_{1}<\cdots<x_{n}=b$. Find the area of the rapezoid under the $<x_{n}=b$ ent joining points $\left(x_{e}, f\left(x_{k}\right)\right)$ and $\left(x_{k+1}, f\left(x_{k+1}\right)\right)$ and $t$ above of Formula ( 2) is the sum of these trapezoidal areas (Figure $7.7 .1) .$

(a) Verify that the average of the left and right end-point approximations as given in Table 7.7 .1 gives Formula ( 2 ) for the trapezoidal approximation.
(b) Suppose that $f$ is a continuous nonnegative function on the interval $(a, b]$ and partition $[a, b]$ with equally spaced points, $a=x_{0}<x_{1}<\cdots<x_{n}=b$. Find the area of the rapezoid under the $<x_{n}=b$ ent
joining points $\left(x_{e}, f\left(x_{k}\right)\right)$ and $\left(x_{k+1}, f\left(x_{k+1}\right)\right)$ and $t$ above of Formula ( 2) is the sum of these trapezoidal areas (Figure $7.7 .1) .$

Key Concepts

Partitioning of the Interval

Dividing the interval of integration into equally spaced subintervals is a key step in numerical methods. This partitioning allows for the systematic application of approximations like the trapezoidal rule on each subinterval, and the sum of these approximated areas yields the total estimated value of the integral.

Trapezoidal Rule

The trapezoidal rule improves upon the basic endpoint approximations by calculating the area under the curve as a series of trapezoids rather than rectangles. By averaging the left and right endpoint function values for each subinterval, the trapezoidal rule offers a more accurate approximation of the integral, which is particularly effective when the function is approximately linear over small intervals.

Area of a Trapezoid

The area of a trapezoid is computed by taking the average of the lengths of the two parallel sides (which correspond to the function values at the endpoints of a subinterval) and multiplying by the distance between them. This geometric interpretation underlies the derivation of the trapezoidal rule for numerical integration.

Numerical Integration

Numerical integration is the process of approximating the value of a definite integral using discrete data points or subintervals. This area of study arises when finding the exact integral is analytically difficult or impossible, and methods such as Riemann sums, the trapezoidal rule, and Simpson’s rule provide practical approximations.

Left and Right Endpoint Approximations

The left endpoint approximation uses the function value at the left side of each subinterval to estimate the area under the curve, while the right endpoint approximation uses the right side value. These methods form simple Riemann sums and provide either an underestimation or overestimation of the integral depending on the monotonicity of the function.

Transcript

00:02 In part a of this problem we are going to show that the average of the left and right end point approximations give rise to formula 2 for the trapezoidal approximation.

00:15 And in part p, we have a function f defined a non -negative on the interval ab, which is subdivided into ends of intervals of the same length by using the points x0 equals a, less than x1 less than up to x n minus 1 less than x sub n which is equal to b.

00:43 Here we want to find the area of the trapezoid under the line segment joining the points x sub k f of x sub k and x sub k plus 1 f of x sub k plus 1 f of x sub k plus 1 and above the interval x of k x of k plus 1 then we will prove that the right side of formula 2 is the sum of these trapezoidal areas so we start in part a by remembering that formula 2 given on chapter 7 .7 of the test book is the trapezoidal approximation t sub n equals b minus a over 2 n times y 0 plus 2 times y sub 1 plus 2 plus 2 plus up 2 times what y sub n minus 1 where y sub k equals f of x of k for k from zero up to n so this is the formula of the trapezoidal approximation and we are going to show in part eight that the average of the left and right end point approximations give rise to this formula too so we start by remembering those and left and right end point approximation so the left end point approximation is given by integral from a to b of f is just equal or approximately equal to b b minus a over n times y0 plus y1 plus up to y sub n minus 1 that is because we are taking the left 10 points, the images of the left end points of the sub -interval, so the last, the first one is y -0 and the last one is y -7 minus 1.

04:00 Then we have the right endpoint approximation, so the integral from a to b is approximately equal to b minus a over n times.

04:28 In this case, we take the right endpoints of the sub -interval, so the first one is y1 then y2 up to y sub n so we can take the average of these two formulas is b minus a over n is one half we can say a half times b minus a over n times y0 plus y1 up to y sub n plus y sub n plus y sub n plus b minus a over n times y1 plus y2 up 2 y sub n that is the sum of these two expressions this one here and this one here over 2 or which is the same and multiply by one half so this is one half times now this term b minus a over n is a common factor we take it out of the square bracket so we get b minus a over n times one half is over 2 n times y 0 plus y1 plus up to n minus 1 plus y1 plus y 2 plus up to y sub n that's equal to b minus a over 2 n times now we see that's equal to b minus a over 2 n times now we see that this term is one time in this whole sum.

06:50 This term appears twice because it's here and here.

06:58 Y2, which is the term that comes here next, appears twice because it's there and there.

07:07 And so on.

07:08 So we are going to have finally that y of n minus 1, which comes before this term here, before this term here, we have y sub n minus 1, which is here...

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