Can someone teach me how to use the trapezoidal rule for integration?

[serussell_]

For instance, how can i apply it to say, *integration symbol* (x^2 - 3)^1/2 with upper limit being 8 and lower being 4??



ps, sorry for the formatting im kinda new and dont know how to work this stuff yet 

[RuiAce]

You haven't specified the number of subintervals.

[serussell_]

5 sub intervals, im not even sure what that means tbh

[RuiAce]

So in this case, we would take the subintervals
- 4 to 4.8
- 4.8 to 5.6
- 5.6 to 6.4
- 6.4 to 7.2
- 7.2 to 8
...because clearly the size of ALL the subintervals is the same; it's 0.8. Note that the size of your original interval was (unsurprisingly) 4, and \( 4\div 5 = 0.8\).

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From here, it's literally a matter of subbing in values into \( f(x) \) (or alternatively drawing up a table of values)

[Never.Give.Up]

Bit late....
but if anyone wants a really basic overview of the trapezoidal rule...here goes....

pretty much find the area of a trapezium
so... the integral between a and b f(x) dx approximately  = 1/2h (a+b)
                                                                                         = 1/2 h (b-a) (f(a) +f(b))
                                                                                         = h/2 (a+b)

you can also use a table which saves a lot of time... and mistakes

E.g. the integral between 4 and 0 x^2 dx
So, y= x^2

x-              0   1   2   3   4
y-              0   1   4   9   16
Wt-              1   2   2   2   1
Wt times y- 0   2   8   18   16

wt= waiting. for trap it is always 121, 1221, 12221, from x^0 to x^n- these are always 1....in between is always 2. You just multiply this by y to get your equation.
Obviously, you get y by subbing it into y=x^2.

So to answer this equation you do...
1/2(0+2+8+18+16)= 22

notice how inside the brackets is the 'wt times y' line,  this combines everything that is in the formula into one simple equation...

lmk if this was really confusing to understand...and i can try showing you another way...but for me this made it soo much quicker- i love integration!!