IB Maths AI SL 5.5 Integration and area: practice questions

(a) Mark scheme

• (M1) evidence of integrating \(\frac{dP}{dx}\), e.g. \(\int (36 - 0.08x)\,dx\);
• A1 \(36x\);
• A1 \(-0.04x^2\) (a constant may be missing at this stage);
• (M1) substituting \(x = 100\) and \(P = 1600\) into their expression with a constant \(c\);
• A1 \(P = -0.04x^2 + 36x - 1600\);

A model answer: \(P = \int (36 - 0.08x)\,dx = 36x - 0.04x^2 + c\). When \(x = 100\): \(1600 = 3600 - 400 + c\), so \(c = -1600\) and \(P = -0.04x^2 + 36x - 1600\).

(b) Mark scheme

• M1 \(\int_{150}^{300} (36 - 0.08x)\,dx\) OR \(P(300) - P(150)\);
• A1 2700 euros;

A model answer: \(P(300) - P(150) = 5600 - 2900 = 2700\) euros (the same as \(\int_{150}^{300} (36 - 0.08x)\,dx\)).