Math · Calculus · Worked example

Accumulate a changing signed rate

Find F(x)=∫₀ˣ(t−1)dt and explain when it decreases.

Find an antiderivative

Integrate each term.

G (t) = \frac{t^{2}}{2} - t

The lower endpoint contributes zero.

F (x) = \frac{x^{2}}{2} - x

F decreases where x−1 is negative, even though x increases.

F^{'} (x) = x - 1

At x=2 the negative and positive areas cancel.

F (2) = 0

F(x)=x²/2−x. It decreases for x<1 and increases for x>1.

Find the slope of F(x)=∫₁ˣt²dt at x=3.

Show the answer and explanation

The Fundamental Theorem gives F′(x)=x²; evaluate at 3.

Change an input and predict the result before checking it.

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.