Integration with a boundary condition
The big idea: Integrating a rate of change gives back the original function PLUS an unknown constant C.
A boundary (initial) condition — one known value of the function — lets you find C and pin down the exact function.
Worked example — find the curve
The gradient of a curve is dy/dx = 6x + 2.
The curve passes through the point (1, 5).
Find y in terms of x.
Step by step
- Integrate the gradient to get y, remembering + C.
- Use the point (1, 5): substitute x = 1, y = 5.
- Solve for C.
Final answer
y = 3x² + 2x.
The method, and a rate-of-change example
Four steps: 1.
Integrate the given rate (derivative) and add + C. 2.
Substitute the known values (the boundary condition). 3.
Solve for C. 4.
Write the full function with the value of C in place.
Worked example — profit from its rate
A company's rate of change of profit is dP/dx = −10x + 460 (in MUR per kg), where x is the number of kg produced.
The profit is 3300 MUR when x = 10.
Find P(x).
Step by step
- Integrate the rate.
- Use the condition P = 3300 when x = 10.
- Simplify and solve for C.
Final answer
P(x) = −5x² + 460x − 800.
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Another context: temperature
Worked example — temperature from its rate
The temperature of a liquid changes at a rate dT/dt = 4t − 12 (°C per minute), where t is the time in minutes.
At t = 0 the temperature is 20 °C.
Find T(t), and the temperature after 5 minutes.
Step by step
- Integrate the rate.
- Use the initial condition T = 20 when t = 0.
- Write the full function, then evaluate at t = 5.
Final answer
T(t) = 2t² − 12t + 20, and T(5) = 10 °C.
Always use the given condition: The constant C is found from the ONE value you are given (the initial/boundary condition).
A solution without using that value — or without + C — loses marks.
Common mistakes
Wrong
- Forgetting the + C when integrating.
- Never using the given point, so C is left unknown.
- Differentiating the rate instead of integrating it.
Right
- Always write + C after integrating.
- Substitute the known value to solve for C.
- Integrate the rate (the reverse of differentiation).
Exam Tips:
- Write the integral and + C before substituting.
- Show the substitution of the boundary condition clearly.
- State the final function with C filled in.