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Definite Integrals
A definite integral is signed accumulation; evaluate an antiderivative at the upper bound minus its value at the lower bound.
How it works
A definite integral is signed accumulation; evaluate an antiderivative at the upper bound minus its value at the lower bound. Name the quantities before calculating, keep each transformation logically equivalent, and check the result against the original conditions.
Worked example
Evaluate ∫₀² x dx.
- An antiderivative of x is x²/2.
- At 2 the value is 2; at 0 it is 0.
- Subtract lower from upper.
The integral is 2.
Try it now
These questions come from the same validated skill generator used by the game.
Common mistakes
- Using a remembered formula without identifying what each symbol represents.
- Changing a value or condition during an intermediate step.
- Skipping the final substitution, estimate, or logical check.
Challenge
Evaluate ∫₁³ 2x dx and interpret it geometrically.
Questions learners ask
What is the key idea in definite integrals?
A definite integral is signed accumulation; evaluate an antiderivative at the upper bound minus its value at the lower bound.
How should I check an answer?
Substitute it into the original conditions, check units and signs, and decide whether its size is reasonable.