Mathematics · 20 / 23
Antiderivatives and integrals: accumulate change
Final-year concepts · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
An antiderivative F of f satisfies F′ = f. Antiderivatives differ by a constant on an interval. A definite integral is F(b) − F(a). For a nonnegative function it is area under the curve; with negative values it is signed area, not total geometric area.
- 01Function f
- 02Antiderivative F
- 03Endpoint values
- 04F(b) minus F(a)
WORKED EXAMPLE
From principle to reasoning
For f(x) = x², use F(x) = x³/3. From 0 to 3 the integral is 9. Differentiating F recovers x².
Geometric area or signed integral?
For f(x)=x on [−1,1], the antiderivative x²/2 gives integral zero: negative and positive parts cancel. Geometric area is 1, from two triangles. It is the integral of |x|, not x.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Integrate 2x from 0 to 2.
One hint
Find a function whose derivative is 2x.
Reveal and understand the solution
F(x) = x², so F(2) − F(0) = 4. The positive graph also forms a triangle of area 4.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Integrate f(x) = 3x² from 0 to 2.
A hint to get started
Find an antiderivative and subtract endpoint values.
Step-by-step solution
- An antiderivative is F(x) = x³.
- F(2) − F(0) = 8 − 0.
- The integral is 8.
Which step did you miss? Retry without help, then explain why the method works.
The integral of x from −1 to 1 is zero. Is its geometric area zero?
A hint to get started
The halves have opposite signs.
Step-by-step solution
- An antiderivative is x²/2, with equal endpoint values.
- The signed integral is zero by cancellation.
- Each triangle has area 1/2, so total geometric area is 1.
Which step did you miss? Retry without help, then explain why the method works.
Find the average value of x² on [0,3].
A hint to get started
Divide the integral by the interval length.
Step-by-step solution
- The integral is [x³/3] from 0 to 3, giving 9.
- Divide by 3 − 0.
- The average is 3, different from the midpoint value 2.25.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Do not confuse derivative and antiderivative or reverse the endpoint subtraction.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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