Physics I / Integrals as accumulation

Practice question · Put in order

Order the steps in constructing a Riemann sum to define the definite integral.

Take the limit as $n \to \infty$ to obtain $\int_a^b f(x) dx$
Form the Riemann sum: Σᵢ f(xᵢ*)· $\Delta x$
Choose a sample point xᵢ* in each subinterval
Divide [a, b] into n equal subintervals of width $\Delta x = (b - a)/n$

Hints

You must chop the interval before you can build rectangles on it.
Taking the limit as the pieces shrink is the final step, not the first.

Show the answer

Divide [a, b] into n equal subintervals of width $\Delta x = (b - a)/n$
Choose a sample point xᵢ* in each subinterval
Form the Riemann sum: Σᵢ f(xᵢ*)· $\Delta x$
Take the limit as $n \to \infty$ to obtain $\int_a^b f(x) dx$

Why

The definite integral is the limit of finer and finer Riemann approximations.

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