IB Maths AI SL 5.5 Integration as anti-differentiation

[Maximum number: 4]

A type of dog food is sold in cylindrical containers. A new label for the container is being considered. The label will be a rectangle that is 5 cm wide and 7 cm high.
The design on the label is a curve, as shown on the following axes, where one unit represents 1 cm for both axes.

Figure for Question IB Maths AI SL 5.5 Integration As Anti Differentiation Questions question 1 — IB Maths AI SL

The values in the table approximate points on the curve, correct to one decimal place.

Table for Question IB Maths AI SL 5.5 Integration As Anti Differentiation Questions question 1 — IB Maths AI SL

(a)

Write down the integral that represents the area of the shaded region.

[ 2 ]

$\quad \int_{0}^{5}\left(-\frac{x^{3}}{3}+3 x^{2}-7 x+6\right) \mathrm{d} x$

Note: Award A2 for an integral with correct integrand and correct limits.

(b)

Hence, find the area of the shaded region.

The dog food is sold in closed cylindrical containers, with radius r and height h.

Figure for Question (b) — IB Maths AI SL

The whole container is made from one type of material, and it is assumed that the thickness of the material is negligible.

Each container has a volume of $400 \mathrm{~cm}^{3}$ .

[ 2 ]

$\quad 15.4\left(15.4166 \ldots, \frac{185}{12}\right) \quad\left(\mathrm{cm}^{2}\right)$

Note: If their limits are outside the domain, award at most A1AOFT.

IB Maths AI SL 5.5 Integration As Anti Differentiation Questions question 1

Question (a)

Question (b)