Fun with Integral Approximation!

Learn how to estimate areas under curves

Example 1: Left-End Method

∫11.5 x dx
Partition points: {1, 1.1, 1.2, 1.3, 1.4, 1.5}
Left-End Rule
We divide the area into 5 rectangles
Each rectangle has width: 0.1
We use the left height of each interval: 1.0, 1.1, 1.2, 1.3, 1.4
Calculate areas: 0.1×1.0 + 0.1×1.1 + 0.1×1.2 + 0.1×1.3 + 0.1×1.4
Total area ≈ 0.1 × (1.0 + 1.1 + 1.2 + 1.3 + 1.4)
Total area ≈ 0.1 × 6.0
Approximate Area: 0.6

Example 2: Right-End Method

∫11.5 x² dx
Partition points: {1, 1.1, 1.2, 1.3, 1.4, 1.5}
Right-End Rule
We divide the area into 5 rectangles
Each rectangle has width: 0.1
We use the right height of each interval: 1.1, 1.2, 1.3, 1.4, 1.5
Calculate areas: 0.1×1.21 + 0.1×1.44 + 0.1×1.69 + 0.1×1.96 + 0.1×2.25
Total area ≈ 0.1 × (1.21 + 1.44 + 1.69 + 1.96 + 2.25)
Total area ≈ 0.1 × 8.55
Approximate Area: 0.855

Example 3: Midpoint Method

∫11.5 (2 - x) dx
Partition points: {1, 1.1, 1.2, 1.3, 1.4, 1.5}
Midpoint Rule
We divide the area into 5 rectangles
Each rectangle has width: 0.1
We use the midpoint height of each interval: 1.05, 1.15, 1.25, 1.35, 1.45
Calculate areas: 0.1×0.95 + 0.1×0.85 + 0.1×0.75 + 0.1×0.65 + 0.1×0.55
Total area ≈ 0.1 × (0.95 + 0.85 + 0.75 + 0.65 + 0.55)
Total area ≈ 0.1 × 3.75
Approximate Area: 0.375