Live paper

Integration and the area under a curve

No calculator

A4 · 10 pp

YCIS · MathematicsNo calculatorIntegration and area under a curve

Yew Chung International School of Hong Kong

Mathematics · Y8

Year 12 · Mathematics AA SL

Integration and the area under a curve

No calculator

name

class

date

Score______ / 90

Functions · Trigonometry and the unit circle · Antiderivatives · Definite integrals · Area

CalculatorNo calculator

Suggested timingPaper 1 50 min Paper 2 40 min

Total marks90 marks (50 + 40)

EquipmentNo calculator. Pen, pencil and ruler only.

FormulaeA clean copy of the AA SL formula booklet may be used.

AccuracyExact answers unless the question asks for a rounded value.

A calculator may not be used. Write all working in the boxes. Exact answers are required unless a question asks otherwise. Marks are awarded for method as well as for the final answer. Write all working in the boxes. Marks are awarded for method as well as for the final answer. A correct answer with no working may not receive full marks. Use radians in every question that mixes trigonometry with calculus. Do not open the markscheme until both sections are finished.

YCIS · Mathematics · Fill-in practice testPage 1

YCIS · MathematicsNo calculatorIntegration and area under a curve

Show all working. Exact answers are required.

[6]

Question 1Indefinite integrals

Find each of the following integrals.

(a)

[2]

$\int (6 x^{2} - 4 x + 7) d x$

(b)

[2]

$\int (3 cos x + 2 sin x) d x$

(c)

[2]

$\int (e^{x} + \frac{1}{x}) d x, x > 0$

[5]

Question 2Definite integral

Evaluate $\int_{1}^{4} (4 x + x^{- 1/2}) d x$ . Give your answer as an integer.

[5]

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YCIS · MathematicsNo calculatorIntegration and area under a curve

[8]

Question 3Unit circle and trigonometric integrals

(a)

[3]

Using the unit circle, write down the exact values of $cos \frac{2 π}{3}, cos \frac{π}{6}, tan \frac{π}{3}$ .

(b)

[3]

Hence, or otherwise, evaluate $\int_{0}^{π /2} (3 sin x + 3 cos x) d x$ .

(c)

[2]

Find $\int cos (3 x) d x$ .

[6]

Question 4Recovering a function from its derivative

The derivative of a function $f$ is given by $f^{'} (x) = 3 x^{2} - 8 x, x \in R$ . The graph of $y = f (x)$ passes through the point $(1, 8)$ .

(a)

[4]

Find $f (x)$ .

(b)

[2]

Find the coordinates of the local maximum point on the graph of $f$ .

YCIS · Mathematics · Fill-in practice testPage 3

YCIS · MathematicsNo calculatorIntegration and area under a curve

[7]

Question 5Area under a quadratic

The curve $y = - x^{2} + 5 x$ intersects the $x$ -axis at the origin $O$ and at the point $A$ .

diagram not to scale

(a)

[2]

Find the coordinates of $A$ .

(b)

[5]

Find the exact area of the region enclosed by the curve and the $x$ -axis.

YCIS · Mathematics · Fill-in practice testPage 4

YCIS · MathematicsNo calculatorIntegration and area under a curve

[9]

Question 6Trigonometric area and the unit circle

Let $f (x) = 3 cos x$ , for $0 \leq x \leq 2 π$ .

diagram not to scale

(a)

[2]

Find the exact $x$ -intercepts of $y = f (x)$ in the interval $0 \leq x \leq 2 π$ .

(b)

[3]

Find the value of $\int_{0}^{2 π} f (x) d x$ and explain, with reference to the graph, why this value is not the geometric area between the curve and the $x$ -axis.

(c)

[4]

Find the total area of the regions enclosed by the curve and the $x$ -axis from $x = 0$ to $x = 2 π$ .

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YCIS · MathematicsNo calculatorIntegration and area under a curve

[9]

Question 7Reverse chain rule and substitution

(a)

[3]

$\int 6 x (x^{2} + 1)^{2} d x$

(b)

[3]

$\int 3 e^{3 x + 1} d x$

(c)

[3]

$\int (4 x + 5)^{- 1} d x$ for $x > - \frac{5}{4}$ .

End of Section A. Check that every indefinite integral includes +C.

YCIS · Mathematics · Fill-in practice testPage 6

YCIS · MathematicsNo calculatorIntegration and area under a curve

A no calculator is required. Unless otherwise stated, give numerical answers correct to three significant figures. Set the calculator to radian mode. Solutions found from a GDC should be supported by suitable working (for example a sketch of the graph used).

[5]

Question 8Numerical definite integral

Use your GDC to evaluate $\int_{1}^{2.5} (e^{- 0.3 x} + ln (x + 1)) d x$ . Give your answer correct to three significant figures.

[5]

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YCIS · MathematicsNo calculatorIntegration and area under a curve

[8]

Question 9Area between two curves

The graphs of $y = 2^{x}$ and $y = 6 - x^{2}$ intersect at the points with $x$ -coordinates $a$ and $b$ , where $a < b$ .

y = 2^xy = 6 − x²

diagram not to scale

(a)

[4]

Use your GDC to find the values of $a$ and of $b$ .

(b)

[4]

Find the area of the region enclosed by the two curves.

YCIS · Mathematics · Fill-in practice testPage 8

YCIS · MathematicsNo calculatorIntegration and area under a curve

[9]

Question 10Kinematics — displacement and distance

A particle moves in a straight line. Its velocity, $v$ m s $^{- 1}$ , at time $t$ seconds is given by $v (t) = 3 cos t + t - 1.2, 0 \leq t \leq 4$ .

diagram not to scale

(a)

[3]

Find the times at which the particle is instantaneously at rest in the interval $0 < t < 4$ .

(b)

[3]

Find the displacement of the particle from $t = 0$ to $t = 4$ .

(c)

[3]

Find the total distance travelled by the particle from $t = 0$ to $t = 4$ .

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YCIS · MathematicsNo calculatorIntegration and area under a curve

[9]

Question 11Trigonometric function, intercepts and area

Consider the function $f (x) = 2 sin (2 x), 0 \leq x \leq π$ .

(a)

[3]

Write down the amplitude, the period and the equation of the principal axis of $y = f (x)$ .

(b)

[3]

The equation $f (x) = 0$ has two solutions in the interval $0 \leq x \leq π$ . Find these solutions.

(c)

[3]

Find the total area of the regions enclosed by the graph of $y = f (x)$ and the $x$ -axis from $x = 0$ to $x = π$ .

[9]

Question 12Functions, composition and a GDC integral

Let $g (x) = ln (x + 2)$ , $x > - 2$ , and let $h (x) = x^{2} + 1$ .

(a)

[2]

Find $(g \circ h) (x)$ .

(b)

[4]

Find the exact value of $\int_{0}^{4} g (x) d x$ .

(c)

[3]

Use your GDC to find the area of the region enclosed by $y = g (x)$ , the $x$ -axis and the lines $x = 0$ and $x = 4$ . Comment on the relationship between your answers to (b) and (c).

End of Section B. Check radian mode was used and that final answers are to 3 s.f.

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