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Integration and Its Applications · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min
A particle moves along a straight line such that its velocity v m s−1v\,\text{m s}^{-1} at time tt seconds is given by v(t)=3t2−4t+5v(t) = 3t^2 - 4t + 5.
(a)
Find an expression for the displacement s(t)s(t), given that s(0)=2s(0) = 2. [3 marks]
(b)
Determine whether the particle is moving away from or towards its initial position at t=3t = 3 seconds. Justify your answer. [2 marks]
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Solutions

2FoundationSAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min
Consider the function f(x)=1x2f(x) = \dfrac{1}{x^2} for x>0x > 0.
(a)
Find ∫f(x) dx\displaystyle\int f(x)\,dx. [2 marks]
(b)
The graph of F(x)F(x), antiderivative of f(x)f(x), passes through the point (1, 3)(1,\,3). Find F(x)F(x). [2 marks]
(c)
Hence find the value of xx for which F(x)=3.5F(x) = 3.5. [1 mark]
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Solutions

3MasterySAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min
A particle moves along a straight line such that its velocity v m s−1v\,\text{m s}^{-1} at time tt seconds is given by v(t)=3t2−12t+9v(t) = 3t^2 - 12t + 9, for t≥0t \geq 0.
(a)
Show that the displacement s(t) ms(t)\,\text{m} of the particle from its starting point is given by s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t. [2 marks]
(b)
Hence find the total distance travelled by the particle in the first 44 seconds. [3 marks]
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Solutions

4ChallengeSAQ-LIndefinite integrals and their propertiess8 marksPaper 1~12 min
A manufacturer produces a decorative rod whose cross-sectional area (in cm2\text{cm}^2) at distance xx metres from one end is given by A(x)=4xe−2x+3A(x) = 4xe^{-2x} + 3, for 0≤x≤30 \leq x \leq 3.
(a)
Determine ∫A(x) dx\displaystyle\int A(x)\,dx. [4 marks]
(b)
Hence find the exact volume of the rod in cm3\text{cm}^3. [2 marks]
(c)
A second rod of the same length has cross-sectional area B(x)=4xe−2x+kB(x) = 4xe^{-2x} + k, where kk is a positive constant. The volume of the second rod is exactly 12 cm312\,\text{cm}^3. Determine the exact value of kk and hence deduce what geometric feature of the two rods differs. [2 marks]
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Solutions

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