Calculus practice

Second Fundamental Theorem worksheets

Differentiate polynomial accumulation functions with constant, variable, and polynomial bounds using exact chain-rule contributions.

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Differentiate each accumulation function F(x). Give an exact polynomial for F'(x).

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Differentiate an accumulation function

Evaluate the integrand at each variable bound, multiply by the bound's derivative, and subtract the lower contribution when the lower bound also varies.

Learning objective

What this activity builds

Apply the Second Fundamental Theorem of Calculus to variable-bound polynomial definite integrals.

Worked example

If F(x) is the integral from 1 to x² of s + 3 ds, what is F'(x)?

2x(x² + 3) = 2x³ + 6x

Evaluate the integrand at x², then multiply by the derivative 2x of the upper bound.

Teaching notes

Make the practice count

A constant lower bound contributes zero.
Differentiate a polynomial upper bound with the Chain Rule.
For a lower bound x, subtract p(x).

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