Riemann sums

Warm-up

Three quick questions from earlier lessons.

Discover

To estimate the area under $y = x^2$ on $[0, 2]$ , you use two rectangles of width 1 whose heights are taken at the left ends, $f(0)$ and $f(1)$ . Is the estimate too small or too big?

Too small
Too big

Key idea

Rectangles approximate area; thinner rectangles do better.

Step 12 rectangles: 2 · 1 + 2 · 2 = 6.
Step 24 rectangles: 1 + 1.25 + 2 + 3.25 = 7.5 — closer.
Step 3More and thinner rectangles approach the exact area, 28/3 ≈ 9.33.

Split

[a, b]

into

n

pieces of width

\Delta x = \dfrac{b - a}{n}

. A Riemann sum adds

f(x_i^*)\,\Delta x

, with

x_i^*

the left end, right end or midpoint of each piece. For increasing

f

, left sums underestimate and right sums overestimate.

Worked example

Right sum for $\displaystyle\int_0^3 x^2\,dx$ with 3 pieces.

\Delta x = 1

Width 1; right ends 1, 2, 3.

1 + 4 + 9 = 14

Heights f(1), f(2), f(3).

R_3 = 14

Compare with the exact value 9: an overestimate.

Your turn, guided

Your turn: the left sum for the same integral, 3 pieces.

L_3 = 5

f(0) + f(1) + f(2).

Practice ★

Estimate $\displaystyle\int_0^{ 5 } x^2\,dx$ with a left Riemann sum of 5 subintervals of width 1.
Estimate $\displaystyle\int_0^{ 6 } x^2\,dx$ with a left Riemann sum of 6 subintervals of width 1.
Estimate $\displaystyle\int_0^{ 4 } x^2\,dx$ with a left Riemann sum of 4 subintervals of width 1.

Watch out!

Practice ★★

Estimate $\displaystyle\int_0^4 (x^2 + 4)\,dx$ with a midpoint sum of 2 subintervals.
Estimate $\displaystyle\int_0^4 (x^2 + 2)\,dx$ with a midpoint sum of 2 subintervals.

$f$ is increasing on $[0, 7]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 7 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact

Exit ticket

Three questions to check you have got it.

Estimate $\displaystyle\int_0^{ 5 } x^2\,dx$ with a left Riemann sum of 5 subintervals of width 1.
Estimate $\displaystyle\int_0^{ 6 } x^2\,dx$ with a left Riemann sum of 6 subintervals of width 1.
Estimate $\displaystyle\int_0^{ 4 } x^2\,dx$ with a left Riemann sum of 4 subintervals of width 1.

Summary card

Riemann sums

$\Delta x = (b - a)/n$ ; add height × $\Delta x$ .
Left, right or midpoint heights.
Increasing $f$ : left under, right over.

Watch out: multiply by the width.

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