Finding a Curve from its Gradient

Integrating a gradient function gives a whole family of curves, $f(x)+c$ . One known point picks out the single curve you want: substitute the point and solve for $c$ .

Edexcel YMA01 WMA11≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Integrate $f'(x)$ to get $f(x)+c$ .
Use a given point to find $c$ .
Use the resulting function to find other values.
Find unknown constants inside the gradient function.

Pause and think

You know your speed at every moment of a journey but not where you started. Can you say exactly where you are now?

Section 2 of 14

Core idea

Start hereWhich curve?

Integrating $\frac{dy}{dx}$ gives a whole family of curves stacked vertically. Tell me one point the curve passes through and exactly one family member fits.

The big idea

Integrate $f'(x)$ to get $f(x)+c$ ; substitute a known point to find $c$ .

1. Fixing the constant

dy/dx = 3x² − 4: only one curve passes through (2, 5)

The whole family

y=x^3-4x+c

has the right gradient; the point

(2,5)

picks out

c=5

Worked example $\frac{dy}{dx}=3x^2-4$ through $(2,5)$

Find the curve.

Goal: $y=f(x)$ .
Strategy: Integrate, substitute.

$y=x^3-4x+c$
Integrate.
$5=8-8+c\Rightarrow c=5$
Point $(2,5)$ .

Answer

y=x^3-4x+5

✓ Check

y(2)=8-8+5=5

✓; derivative

3x^2-4

✓.

Worked example $f'(x)=\frac6{\sqrt x}$ , $f(4)=30$

Find $f(x)$ .

Goal: Function.
Strategy: $6x^{-1/2}$ .

$f(x)=\frac{6x^{1/2}}{1/2}+c=12\sqrt x+c$
Integrate.
$24+c=30\Rightarrow c=6$
$12\sqrt4=24$ .

Answer

f(x)=12\sqrt x+6

✓ Check

f(4)=30

✓.

Common misconceptionForgetting c Without

c

you cannot use the point, and the answer is wrong.

Pause and think

$\frac{dy}{dx}=2x$ , curve through $(1,4)$ ?

Summary

Integrate, then use the point.

Particular solution: The one curve fitting a given point.

HistoryThe fundamental theorem. Isaac Barrow, Newton's teacher at Cambridge, was among the first to see that finding areas and finding tangents are inverse processes. Newton and Leibniz turned that insight into the fundamental theorem of calculus.

Fun factReconstructing a flight path. If a plane's black-box recorder logs its rate of climb, investigators can integrate it to reconstruct the plane's height over time, provided they know its height at one moment (usually take-off).

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