A-Level H2 Math | 5 Essential Questions
Applications of Integration
Integration is a very useful extension of differentiation and relatively easy to understand. They are also known as the opposite of the derivatives. The definite integral is a powerful concept used to represent the area under a curve, which turns out to be useful in a lot of calculations.Â
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2013 RVHS Promo Q3
The diagram shows the sketch of the curve $C$, ${{\left( y-1 \right)}^{2}}=x\sqrt{{{x}^{2}}-1}$, with the vertex at $\left( 1,1 \right)$.
(i)
Write down the equation of the graph when $C$ is translated $1$ unit in the negative $y$-direction.
[1]
(i) Write down the equation of the graph when $C$ is translated $1$ unit in the negative $y$-direction.
[1]
(ii)
The shaded region $R$, bounded by $C$ and the vertical line, $x=a$, is rotated through $\pi $ radians about the line $y=1$. By using the substitution $u=\sqrt{{{x}^{2}}-1}$, or otherwise, find the exact volume obtained in terms of $a$.
[5]
(ii) The shaded region $R$, bounded by $C$ and the vertical line, $x=a$, is rotated through $\pi $ radians about the line $y=1$. By using the substitution $u=\sqrt{{{x}^{2}}-1}$, or otherwise, find the exact volume obtained in terms of $a$.
[5]
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2013 AJC Promo Q8 (b)
The diagram shows a shaded region $R$ bounded by the curve ${{\left( y-2 \right)}^{2}}=x+1$ and the line $y+2x=6$.
[4]
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2017 YJC P1 Q9
(a)
By using the substitution $x=3\sec \theta $, evaluate $\int_{3\sqrt{2}}^{6}{\frac{3x+1}{\sqrt{{{x}^{2}}-9}}\text{d}x}$ exactly.
[5]
(a) By using the substitution $x=3\sec \theta $, evaluate $\int_{3\sqrt{2}}^{6}{\frac{3x+1}{\sqrt{{{x}^{2}}-9}}\text{d}x}$ exactly.
[5]
(b)
The diagram shows an ellipse with equation $\frac{{{x}^{2}}}{16}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1$. (b)
The diagram shows an ellipse with equation $\frac{{{x}^{2}}}{16}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1$.
(i)
Find the area of the shaded region, giving your answer correct to $3$ decimal places.
[2]
[2]
(ii)
Find the exact volume of the solid generated when the shaded region is rotated $180{}^\circ $ about the $y$-axis.
[4]
(ii) Find the exact volume of the solid generated when the shaded region is rotated $180{}^\circ $ about the $y$-axis.
[4]
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2020 ASRJC P2 Q4
(a)
Find $\int{\left( {{\cot }^{6}}2x+{{\cot }^{4}}2x-\sin \frac{x}{2}\sin \frac{3x}{2} \right)}\text{ d}x$.
[3]
(a) Find $\int{\left( {{\cot }^{6}}2x+{{\cot }^{4}}2x-\sin \frac{x}{2}\sin \frac{3x}{2} \right)}\text{ d}x$.
[3]
(b)
$R$ is the finite region bounded by the two curves $y={{\text{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.
Find the volume of the solid formed when $R$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.
[3]
(b) The diagram shows the graphs of $y={{\text{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.
$R$ is the finite region bounded by the two curves $y={{\text{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.
Find the volume of the solid formed when $R$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.
[3]
(c)
$x=2\left( \sin \frac{t}{2}-\cos \frac{t}{2} \right)$, $y=\sin \frac{t}{2}+\cos \frac{t}{2}$, for $-\frac{3\pi }{2}\le t\le -\frac{\pi }{2}$.
The line $L$ with equation $y=-2x-5$ meets the curve $C$ at the point $\left( -2,\,-1 \right)$. $S$ is the region enclosed by line $L$, curve $C$and the $x$-axis as shown in the diagram below.
(c) A curve $C$ has parametric equations
$x=2\left( \sin \frac{t}{2}-\cos \frac{t}{2} \right)$, $y=\sin \frac{t}{2}+\cos \frac{t}{2}$, for $-\frac{3\pi }{2}\le t\le -\frac{\pi }{2}$.
The line $L$ with equation $y=-2x-5$ meets the curve $C$ at the point $\left( -2,\,-1 \right)$.
$S$ is the region enclosed by line $L$, curve $C$and the $x$-axis as shown in the diagram below.
(i)
Show that the x-intercept of curve $C$ is $-2\sqrt{2}$.
[2]
(i) Show that the x-intercept of curve $C$ is $-2\sqrt{2}$.
[2]
(ii)
Find the exact area of $S$.
[6]
(ii) Find the exact area of $S$.
[6]
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2011 CJC P1 Q8
The diagram below shows the curve with equation $y={{x}^{2}}-1$, $x>0$ and the tangent $y=2x-2$ to the curve at the point $(1,0)$.
(i)
Write down the coordinates of the point where this tangent meets the line $x=3$.
[1]
(i) Write down the coordinates of the point where this tangent meets the line $x=3$.
[1]
The region, $R$, shaded in the diagram, is bounded by the curve, the tangent to the curve at $(1,0)$ and the line $x=3$.
Find
(ii)
the area of $R$,
[2]
(ii) the area of $R$,
[2]
(iii)
the exact volume of the solid of revolution obtained when $R$ is rotated through $2\pi $ radians about the $y$-axis.
[4]
(iii) the exact volume of the solid of revolution obtained when $R$ is rotated through $2\pi $ radians about the $y$-axis.
[4]
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A curve $C$ has equation $y=\frac{1}{\sqrt{4-{{x}^{2}}}}$ for $-1\le x\le 1$. The region $R$ is enclosed by $C$, the $x$- axis and the lines $x=-1$ and $x=1$.
(i)
Find the exact value of the area of $R$.
[3]
(ii)
Find the exact value of the volume generated when $R$ is rotated through two right angles about the $x$- axis.
[3]
(iii)
Show that the volume generated when $R$ is rotated through two right angles about the $y$- axis is $\pi \,(4-2\sqrt{3})$.
[4]
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