The trapezium rule

Objective. Approximate definite integrals and discuss error.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

The trapezium rule approximates a definite integral by dividing the area under a curve into equal-width strips and treating each strip as a trapezium. With n strips of width h = (b−a)/n and y-values y₀,...,yₙ, the estimate is ∫[a to b] y dx ≈ (h/2)[y₀ + yₙ + 2(y₁+...+yₙ₋₁)]. For y = x² between x = 0 and x = 2 with 2 strips, h = 1, giving (1/2)[0 + 4 + 2(1)] = 3. The rule tends to overestimate for convex curves and underestimate for concave ones.

Worked example

Use the trapezium rule with 4 strips to estimate ∫₀⁴ √x dx, given y-values at x = 0,1,2,3,4 of 0, 1, 1.414, 1.732, 2.

1 $Identify strip width h = (4-0)/4 = 1$
2 $Sum the end ordinates: y₀ + y₄ = 0 + 2 = 2$
3 $Sum and double the interior ordinates: 2(1 + 1.414 + 1.732) = 8.292$
4 $Apply (h/2)[ends + interior] = (1/2)(10.292) = 5.146$

Answer: 5.146 (3 d.p.)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common error is doubling every y-value, including the first and last ordinates, instead of only doubling the interior ones. Always double only y₁ through yₙ₋₁, keeping y₀ and yₙ un-doubled.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard