Study guides › Level 10 · Calculus

Area under a curve

Level 10 · Calculus · for questions like “What is the area under y = x² from x = 0 to x = 3?”

💡 The idea

The area under a curve between two x’s is found with an antiderivative F:

area = F(end) − F(start)

✏️ Worked examples

1. y = x² from 0 to 3: F(x) = x³/3. F(3) − F(0) = 27/3 = 9.

2. y = 2x from 0 to 5: F(x) = x². 25 − 0 = 25. Check: it's a triangle, ½ × 5 × 10 = 25. ✓

3. y = 3x² from 0 to 2: F(x) = x³, so 8.

⚠️ Watch out

• Take away F(start). It's 0 when you start at 0, but not otherwise.

🎯 Try these

1. y = 4x from 0 to 3
2. y = 3x² from 0 to 4
3. y = x² from 0 to 6

Answers: 1) 18 2) 64 3) 72

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.