Area of shortest rectangle
= Width × height
= 1/n × 2⁰
= 1/n × 1

(because when x = 0, y = 2⁰ = 1. You use the left side of each rectangle that touches the graph to determine its height)

Area of second shortest rectangle = 1/n × 2¹/ⁿ
Area of third rectangle = 1/n × 2²/ⁿ
Area of fourth rectangle = 1/n × 2³/ⁿ
...
Area of tallest rectangle = 1/n × 2⁽ⁿ⁻¹⁾/ⁿ


Notice the common factor 1/n so we can factor this out (distributive law of multiplication)

Total area of all rectangles

= 1/n (1 + 2¹/ⁿ + 2²/ⁿ + 2³/ⁿ + ... + 2⁽ⁿ⁻¹⁾/ⁿ )

= 1/n (1 + 2¹/ⁿ + (2¹/ⁿ)² + (2¹/ⁿ)³ + ... + (2¹/ⁿ)ⁿ⁻¹ )

= 1/n (1 - (2¹/ⁿ)⁽ⁿ⁻¹⁾⁺¹ ) / (1 - 2¹/ⁿ)

Here we are taking x = 2¹/ⁿ , n is replaced by n - 1, and we apply the given equation.

= 1/n (1 - (2¹/ⁿ)ⁿ ) / (1 - 2¹/ⁿ)

= 1/n (1 - 2¹) / (1 - 2¹/ⁿ)

= 1/n (-1)/(1 - 2¹/ⁿ)

= 1/n (1/(2¹/ⁿ - 1))

(Shown)

As n → ∞, the number of rectangles tends to infinity as well.

This means the width of each rectangle becomes smaller and smaller and so small and tends to 0.

Basically , the entire area under the curve is covered with no gaps.

So this is just integrating y = 2ˣ (area under the curve) from x = 0 to x = 1 (recall that n/n = 1)

∫₀¹ 2ˣ dx

= [1/ln2 (2ˣ)]₀¹

= [1/ln2 (2¹)] - [1/ln2 (2⁰)]

= 2/ln2 - 1/ln2

= 1/ln2