Spell of Linear Enchantment
∫01 (5x + 4) dx
✧ Step 1: Summon the Limit Definition
∫ab f(x) dx = limn→∞ Σr=1n f(a + r·h)·h
Where h = (b-a)/n is the magical width of each partition
✧ Step 2: Prepare the Ingredients
For our spell: a = 0, b = 1, f(x) = 5x + 4
h = (1-0)/n = 1/n
xr = 0 + r·h = r/n
✧ Step 3: Cast the Summation Spell
Σ f(xr)·h = Σ (5(r/n) + 4)·(1/n)
= (5/n²)Σ r + (4/n)Σ 1
✧ Step 4: Invoke Summation Formulas
= (5/n²)·n(n+1)/2 + (4/n)·n
= 5(n+1)/(2n) + 4
✧ Step 5: Release the Limit Magic
limn→∞ [5(n+1)/(2n) + 4] = 5/2 + 4 = 13/2
Click "Visualize the Spell" to begin magic
The magical result is: 13/2 or 6.5
Spell of Quadratic Sorcery
∫12 (4x² - 1) dx
✧ Step 1: Summon the Limit Definition
∫ab f(x) dx = limn→∞ Σr=1n f(a + r·h)·h
Where h = (b-a)/n is the magical width of each partition
✧ Step 2: Prepare the Ingredients
For our spell: a = 1, b = 2, f(x) = 4x² - 1
h = (2-1)/n = 1/n
xr = 1 + r·h = 1 + r/n
✧ Step 3: Cast the Summation Spell
Σ f(xr)·h = Σ [4(1 + r/n)² - 1]·(1/n)
= Σ [4(1 + 2r/n + r²/n²) - 1]·(1/n)
✧ Step 4: Expand the Magic
= Σ [3 + 8r/n + 4r²/n²]·(1/n)
= (3/n)Σ1 + (8/n²)Σr + (4/n³)Σr²
✧ Step 5: Invoke Summation Formulas
= (3/n)·n + (8/n²)·n(n+1)/2 + (4/n³)·n(n+1)(2n+1)/6
= 3 + 4(n+1)/n + [2(n+1)(2n+1)]/(3n²)
✧ Step 6: Release the Limit Magic
limn→∞ [3 + 4(1 + 1/n) + (2(1 + 1/n)(2 + 1/n))/3] = 3 + 4 + 4/3 = 25/3
Click "Visualize the Spell" to begin magic
The magical result is: 25/3 or ≈8.333