What is the Fundamental Theorem of Calculus?

It links the two kinds of integral: the definite integral of f from a to b equals F(b) − F(a), where F is any function whose derivative is f.

Does DSE M1 include the area between two curves?

No. M1 covers only the area bounded by a curve, the x-axis and two vertical lines.

What is a dummy variable?

The letter used inside a definite integral does not affect its value: the integral of f(x) dx from a to b equals the integral of f(t) dt from a to b.

Definite Integration and Area Under a Curve (DSE M1)

A definite integral is the limit of a sum of areas of rectangles under a curve, evaluated as ∫_a^b f(x) dx = F(b) − F(a). In M1 it gives the area bounded by y = f(x), the x-axis and the lines x = a and x = b.

Solve Definite Integration and Area questions with AI

The area under y = e^(x/2) from Example 1, growing as the upper limit moves from 0 to 3, where it reaches about 6.963. Try it yourself →

Properties of Definite Integrals

Fundamental Theorem∫_a^b f(x) dx = F(b) − F(a)

Reversing limits∫_a^b f(x) dx = −∫_b^a f(x) dx

Splitting the interval∫_a^c = ∫_a^b + ∫_b^c

Area∫_a^b f(x) dx where f(x) ≥ 0

Worked Examples — Definite Integration and Area

Example 1: Area under an exponential curve

Find the area bounded by y = e^x/2, the x-axis and the lines x = 0 and x = 3.

∫₀³ e^x/2 dx = [2e^x/2]₀³
= 2e^1.5 − 2
≈ 6.963

Answer: 2(e^1.5 − 1) ≈ 6.963 square units

Example 2: Substitution with a change of limits

Evaluate ∫₀¹ x(x² + 1)³ dx.

Let u = x² + 1, du = 2x dx
Limits: x = 0 → u = 1, x = 1 → u = 2
∫₁² ½u³ du = [u⁴/8]₁² = 16/8 − 1/8
= 15/8

Answer: 15/8

💡 Exam Traps & Secrets

⚠️ Change the limits with the variableAfter u = g(x), the limits become g(a) and g(b). The curriculum notes state this explicitly; integrating in u with x-limits is wrong.

⚠️ What M1 areas do not includeThe area between a curve and the y-axis, and the area between two curves, are not required in M1. Volumes of revolution belong to M2.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Area)Find the area under y = 3x² + 2 from x = 1 to x = 3 and shade it.

(Substitution with limits)Evaluate the integral of 2x e^(x²) from 0 to 1.

(Rectilinear motion)A particle has velocity v = 6e^(−0.5t). Find the distance travelled from t = 0 to t = 4.