> divide the interval into two parts: [a,(a + b/2)] and [(a + b/2),b];
Try: [a,(a+b)/2 and [(a+b)/2, b]
(a+b)/2 is the mid-point between a and b: (a + b/2) ain't.
-- Dan
> dave yrueta wrote:
>
> Hi All --
>
> Am sending out a message in a virtual bottle for help with HtDP
> exercise 23.4.1, reproduced below:
>
> A general integration function must consume three inputs: a, b, and
> the function f. The fourth part, the x axis, is implied. This suggests
> the following contract:
>
> ;; integrate : (number -> number) number number -> number
> ;; to compute the area under the graph of f between a and b
> (define (integrate f a b) ...)
> Kepler suggested one simple integration method. It consists of three
> steps:
>
> divide the interval into two parts: [a,(a + b/2)] and [(a + b/2),b];
>
> compute the area of each trapezoid; and
>
> add the two areas to get an estimate at the integral.
>
> Exercise 23.4.1. Develop the function integrate-kepler. It computes
> the area under some the graph of some function f between left and
> right using Kepler's rule.
>
> Specifically, I'm stuck at the "create examples" part of the design
> recipe, because I don't understand the "integration method" outlined
> by the exercise.
>
> First, what does it mean to "divide the interval into two parts?"
> Suppose I substitute values for variables a and b (a=3 and b=6). I
> end up with a pair of values: [3, 6] and [6, 6]. Are these x,y
> coordinates for points on a graph? If so, how do they relate to the
> second step of the integration method, which is to determine the area
> for "each trapezoid?"
>
> I know the area for a trapezoid = a (b1 + b2 /2) where a is the
> altitude, and b1 and b2 are the length of the two bases. My problem
> is deriving values for a, b1 and b2 from the integrate-kepler function
> arguments.
>
> Finally, I have no idea how argument f, the line function, fits into
> all this. I don't even know what a function for a line looks like (y
> = x + 10?)
>
> HtDP supplies an illustration to help visualize the problem (http://
> www.htdp.org/2003-09-26/Book/curriculum-Z-H-29.html#node_sec_23.4). I
> haven't the faintest idea how to interpret it:
>
> I'm sure someone even casually acquainted with basic calculus could
> solve this problem easily. Unfortunately, I'm relatively math-
> illiterate, so figuring out how all the parts fit together has eluded
> me. I did manage to solve the other mathematical examples which
> appeared earlier in the chapter because the mechanics of determining
> the solution appeared clear to me. Not so here. I'm hoping that if
> someone can help me formulate some examples, I'd be on my way.
>
> Thanks!
>
> Dave Yrueta
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