Edexcel IAL Mathematics P2.8.2 interpreting definite integrals

Area questions require more than evaluating an integral: identify the shaded region, choose the correct limits and subtract line, curve or axis contributions.

Exam points

Set up the shaded area using curve equations, straight lines, limits and intersections.
Integrate curve differences or subtract triangle and rectangle areas where required.
Give exact or rounded areas from algebraic integration without calculator-only shortcuts.

[Maximum number: 3]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Figure 3

Figure 3 shows
- the curve $C_{1}$ with equation $y=x^{3}-5 x^{2}+3 x+14$
- the circle $C_{2}$ with centre T

The point T is the minimum turning point of $C_{1}$
Using Figure 3 and calculus,

Find the exact area of R.

$\int_0^2 (x^3-5x^2+3x+14)\,dx =\left[\frac14x^4-\frac53x^3+\frac32x^2+14x\right]_0^2$
The triangular area to subtract is
$\frac12\left(\frac{22}{3}+8\right)2=\frac{46}{3}.$
Hence
$\text{Area }R=\frac{74}{3}-\frac{46}{3}=\frac{28}{3}.$
M1 dM1 A1

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P2.8.2 - Interpretation of the definite question 1