Integration Formulas and the Net Change Theorem

Extending the formula table

The net change theorem

Worked applications

The substitution preview

The integral of x (x^2 + 1)^3 dx evaluates in one line to (x^2 + 1)^4/8 + C. Which pattern made this immediate?

Which integral has the form u prime over u and therefore evaluates to a logarithm?

The net change theorem expresses the total change of a quantity Q with rate Q prime over [a, b] as what?

A particle's velocity is negative on part of [a, b]. How does the displacement integral differ from the distance traveled?

In applications, which direction does the net change theorem most commonly enable?

A factory's marginal cost is C prime of q = 5 + 0.2q dollars per unit. The cost of increasing production from q = 0 to q = 10 is the integral of C prime, which equals what number? Answer with a number.

A flow rate is measured in liters per minute and integrated over a time span in minutes. According to the net change theorem, what units must the resulting integral carry?

Velocity v(t) = 2t on [0, 3] gives displacement the integral of 2t dt. What displacement results? Answer with a number.

To compute distance traveled from a velocity that changes sign once on [a, b], the integral of the absolute value must be split. Into how many pieces? Answer with a number.

Water flows into a tank at rate r(t) = 20 - 0.5t liters per minute for t from 0 to 20 minutes. What total volume flows in? Answer with a number.