IB Maths AA SL 5.11 Definite Integrals and Areas Questions

Practise IB Mathematics SL 5.11 by applying definite integrals and areas methods to questions from the Question Bank.

Exam points

Evaluate ∫ₐᵇ g′(x) dx as g(b) − g(a), writing the correct definite-integral expression before calculating.
Find areas under or between curves by accounting for the sign of the integrand and splitting at intersections or axis crossings where necessary.

[Maximum number: 3]

Consider the function $f(x)=\frac{2}{\pi} \sin (3 \pi x)+2$ , where $0 \leq x \leq 2$ . The following diagram shows the graph of f.

Figure for Question IB Maths AA SL 5.11 Definite Integrals and Areas Questions question 1 — IB Maths AA SL

Find the area of the shaded region.

(f) EITHER

Attempt to form the required integral involving subtraction in either order.
$\int_{t_{1}}^{t_{2}}\left|L_{1}-f(x)\right| \mathrm{d} x$
correct expression with limits 1 and their x value at B
$\int_{1}^{1.64856 \ldots}\left|\left(\frac{1}{6} x+\frac{11}{6}\right)-\left(\frac{2}{\pi} \sin 3 \pi x+2\right)\right| \mathrm{d} x$ (or equivalent)

OR
intersection point between A and B is ( 1.34287 . . ., 2.05714 . .. )
subtracting $L_{1}$ and f, in either order, and substituting into area of region formula for one area with correct limits.

Two correct expressions OR correct values
$\int_{1}^{1.34287 \ldots}\left(\left(\frac{1}{6} x+\frac{11}{6}\right)-\left(\frac{2}{\pi} \sin 3 \pi x+2\right)\right) \mathrm{d} x=(0.144618 \ldots)$ AND
$\int_{1.34287 \ldots}^{1.64856 \ldots}\left(\left(\frac{2}{\pi} \sin 3 \pi x+2\right)-\left(\frac{1}{6} x+\frac{11}{6}\right)\right) \mathrm{d} x=(0.108585 \ldots)$

THEN
0.253204...
(area =) 0.253

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IB Maths AA SL 5.11 Definite Integrals and Areas Questions question 1