Integration: accumulation and area
Reverse differentiation and interpret a definite integral carefully.
Before you begin
- Differentiation of polynomials
- Signed areas and graphs
Find an antiderivative
An antiderivative differentiates back to the integrand. For xⁿ with n ≠ −1, increase the exponent by one and divide by that new exponent. Include a constant C for an indefinite integral because every constant differentiates to zero. Check the result by differentiating it. The x⁻¹ case needs a logarithm and is excluded from this power rule.
Evaluate signed accumulation
For an antiderivative F, the definite integral from a to b is F(b) − F(a). Areas below the horizontal axis count negatively, so a definite integral may differ from total geometric area. In a rate model, it gives net accumulated change. Specify bounds and units before interpreting the numerical result.
Worked example
Evaluate the integral of 2x from x = 1 to x = 3.
Show the worked solution
- An antiderivative of 2x is x².
- Evaluate at the upper bound: 3² = 9.
- Subtract the value at the lower bound: 9 − 1² = 8.
8.
Try it yourself
Which is an antiderivative of 6x²?
Common mistakes
- Net signed area is not always total area.
What can you explain now?
A velocity is v(t) = 3t² m/s for 0 ≤ t ≤ 2 s. Find the displacement over that interval.
Compare with the explanation
8 m.
Integrate 3t² to t³ and evaluate 2³ − 0³ = 8. Velocity is nonnegative here, so displacement and distance travelled have the same numerical value.
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Common Core mathematics standards ↗