Question

trapezoidal rule n=6 integral upper integral pi lower 0 {cos(x/2) dx

Answer 1

\[\int\limits_{0}^{\pi}\cos (x/2) dx\]

Answer 2

if you want to use the trapezoidal rule, all you are trying to do is to
estimate the area under the curve cos(x/2) using 6 trapezoids.

Answer 3

so that each height of the trapezoids are pi/6

Answer 4

since the area of a trapezoid can be calculated by the formula

1/2(b_1+b_2)h,

all you have to do now is to find out b_1 and b_2 in each
trapezoid.

Answer 5

for the left most trapezoid, b_1 will be the left base,
which can be calculated by cos(0/2) = 1.
b_2 would be the right base an similarly can be calculated by
cos(pi/6 /2) = not a nice number.
the height of the trapezoid was h = pi/6, so the area is
1/2 ( 1 + cos(pi/12))*pi/6.

Answer 6

let's do one more step

Answer 7

the next trapezoid will have b_1 which is
cos(pi/12).
As you noticed, it is the same as the b_2 for the first trapezoid.
They have to be the same because they share that side.

Answer 8

Anyway, b_2 for the second trapezoid would be
calculated by
cos( 2*pi/6 /2) because now the x value you are
plugging in is 2*pi/6.

Answer 9

the height is the usual, pi/6
so the area of the second trapezoid would be

1/2 (cos(pi/12) + cos(pi/6)) * pi/6

Answer 10

you keep doing this until you get the 6 trapezoids :)

Answer 11

thanks

Answer 12

to do simpsons rule same problem it will be aprox the same answer correct?

Answer 13

yes, but instead of a trapezoid, now you will be using a quadratic.
good luck :)