Methods for Estimating Integrals without Calculators

[intenzxboi]

Homework Statement

estimate int 1/ln x between 2 and 4


well i thought that for this to work you would have to integrate first and then find f(4) - f(2)

but how come right now we are learning about using the midpoint, left, right, trap and simpson rule.

but instead of integrating it first they are just using the equation without integrating first??

also for int 1/ln x between 2 and 4 using one of the estimating methods(ex. right) is there any way to do this problem without a calculator??

 

[Mark44]

Well, good luck on finding an antiderivative for 1/(ln x)!

Otherwise, you're going to be stuck with calculating an estimate for the integral, rather than an exact value.

You'll need a calculator to do your estimate.

 

[csprof2000]

You can approximate it by hand.

1. Find the the first, second, and possibly third derivatives.
2. Evaluate at x = 2. Call them a = f ''(2), b = f '(2), c = f (2)
3. Then call 1/ln(x) = c + b(x-2) + a(x-2)^2.

Then integrate the result from step 3, and that will be a good approximation.

 

[Mark44]

You can approximate it by hand.

1. Find the the first, second, and possibly third derivatives.
2. Evaluate at x = 2. Call them a = f ''(2), b = f '(2), c = f (2)
3. Then call 1/ln(x) = c + b(x-2) + a(x-2)^2.

Then integrate the result from step 3, and that will be a good approximation.

Sure, you don't need a calculator for that estimate, but if the OP doesn't know about Taylor series, this will be pure magic.

 

[yyat]

Homework Statement

estimate int 1/ln x between 2 and 4well i thought that for this to work you would have to integrate first and then find f(4) - f(2)

but how come right now we are learning about using the midpoint, left, right, trap and simpson rule.

but instead of integrating it first they are just using the equation without integrating first??

also for int 1/ln x between 2 and 4 using one of the estimating methods(ex. right) is there any way to do this problem without a calculator??

Since you mentioned http://en.wikipedia.org/wiki/Simpson_rule" this may be the way you are supposed to do it. Note that the formula does not involve taking the antiderivative of 1/ln x, you only need the values of that function at 2,3,4.

 

[intenzxboi]

ic ic thanks