Question

Express the integral as a limit of Riemann sums?

`\int_{4}^{12}[\ln(1+x^2)-\sin(x)]dx`
Do not evaluate the limit.

Please don't use anything above Calculus I level.

Answer 1

Here is a limit definition of the definite integral. (I'd guess it's the one you are using.)

.`int_a^b f(x) dx = lim_(nrarroo) sum_(i=1)^n f(x_i)Deltax`.

Where, for each positive integer `n`, we let `Deltax = (b-a)/n`

And for `i=1,2,3, . . . ,n`, we let `x_i = a+iDeltax`. (These `x_i` are the right endpoints of the subintervals.)

Let's go one small step at a time.

`int_4^12 [ln(1+x^2)-sinx] dx`.

Find `Delta x`

For each `n`, we get

`Deltax = (b-a)/n = (12-4)/n = 8/n`

Find `x_i`

And `x_i = a+iDeltax = 4+i8/n = 4+(8i)/n`

Find `f(x_i)`

`f(x_i) = ln(1+x_i""^2)-sinx_i = ln(1+(4+(8i)/n)^2)-sin(4+(8i)/n)`

`int_4^12 [ln(1+x^2)-sinx] dx`

`= lim_(nrarroo)sum_(i=1)^n [ (ln(1+(4+(8i)/n)^2)-sin(4+(8i)/n))8/n]`.

Expand/simplify `(1+(4+(8i)/n)^2)` if required to `(17+(64i)/n + (64i^2)/n^2)`