Signed integral and total area

Practise signed integral and total area with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

Let $f(x)=2x-2$ for $0\leq x\leq3$. (a) Evaluate $\int_0^3 f(x)\,dx$. [2 marks] (b) Find the total area between the graph and the x-axis. [2 marks]

Worked explanation

• Part (a)

  An antiderivative is $x^2-2x$. The integral is $[x^2-2x]_0^3=9-6=3$.

• Part (b)

  The sign changes at $x=1$. Total area is $-\int_0^1(2x-2)\,dx+\int_1^3(2x-2)\,dx=1+4=5$.

(a) $3$.; (b) $5$ square units.

Original VectorPaths practice; not an official VCAA question or marking rubric.