6h • General discussion
Worked Solution — Problem of the Week #2 ✅
Here's the full worked solution to this week's problem!
Problem: Find the exact area enclosed between f(x) = x² - 4 and the x-axis.
Step 1 — Find the x-intercepts:
x² - 4 = 0 → (x - 2)(x + 2) = 0 → x = -2 or x = 2
Step 2 — Check whether the curve is above or below the x-axis on [-2, 2]:
f(0) = -4 < 0 — the curve is below the x-axis on [-2, 2]
Step 3 — Set up the integral and take the absolute value:
Area = |∫ from -2 to 2 of (x² - 4) dx|
Step 4 — Evaluate:
= |[x³/3 - 4x] from -2 to 2|
= |(8/3 - 8) - (-8/3 + 8)|
= |8/3 - 8 + 8/3 - 8|
= |16/3 - 16|
= |-32/3|
= 32/3 units²
The exact area is 32/3 square units. ✅
Common mistake to avoid: always check whether the curve is above or below the x-axis before integrating. If you forgot the absolute value here you would get -32/3 — which is the signed area, not the geometric area.
New Problem of the Week drops Monday! 💪
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Worked Solution — Problem of the Week #2 ✅
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