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PSLE

PSLE Foundation Mathematics

The curve \(y=\dfrac{25}{(x+2)^2}-1\) meets the positive \(x\)-axis at \(A\). The shaded region \(P\) is bounded by the curve and the coordinate axes. The shaded region \(Q\) is bounded by the curve, the \(x\)-axis and the line \(x=8\).
(i) Find the coordinates of \(A\).
(ii) Find the areas of \(P\) and \(Q\).
(iii) Find the ratio of the area of \(P\) to the area of \(Q\).

The curve \(y=\dfrac{25}{(x+2)^2}-1\) meets the positive \(x\)-axis at \(A\). The shaded region \(P\) is bounded by the curve and the coordinate axes. The shaded region \(Q\) is bounded by the curve, the \(x\)-axis and the line \(x=8\).

Find the ratio of the area of \(P\) to the area of \(Q\).

Area Enclosed by the Curve and the X-Axis

Rational curve with positive and negative regions

Integration

<p>The curve \(y=\dfrac{18}{(x+1)^2}-1\) cuts the positive \(x\)-axis at \(A\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q12-parts-item-1" data-list-item-path="amarea-q12-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(A\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q12-parts-item-2" data-list-item-path="amarea-q12-parts-item-2"><div class="list-item-paragraph">Find the area of the shaded region bounded by the curve, the coordinate axes and the line \(x=2\).<span class="inline-marks">[4]</span></div></li></ol>

The curve \(y=\dfrac{18}{(x+1)^2}-1\) cuts the positive \(x\)-axis at \(A\).

Find the area of the shaded region bounded by the curve, the coordinate axes and the line \(x=2\).

Shifted inverse-square curve above the x-axis

<p>The curve \(y=(x-2)(x-5)\) intersects the coordinate axes at \(A\), \(B\) and \(C\), as shown. The shaded regions are denoted by \(P\) and \(Q\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q26-parts-item-1" data-list-item-path="amarea-q26-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(A\), \(B\) and \(C\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q26-parts-item-2" data-list-item-path="amarea-q26-parts-item-2"><div class="list-item-paragraph">Find the areas of \(P\) and \(Q\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="amarea-q26-parts-item-3" data-list-item-path="amarea-q26-parts-item-3"><div class="list-item-paragraph">Hence find the ratio of the area of \(P\) to the area of \(Q\).<span class="inline-marks">[1]</span></div></li></ol>

The curve \(y=(x-2)(x-5)\) intersects the coordinate axes at \(A\), \(B\) and \(C\), as shown. The shaded regions are denoted by \(P\) and \(Q\).

Find the coordinates of \(A\), \(B\) and \(C\).

Hence find the ratio of the area of \(P\) to the area of \(Q\).

Unequal regions on opposite sides of the x-axis

<p>The diagram shows part of the curve \(y=\dfrac{4-x}{\sqrt{x+1}}\), which meets the positive \(x\)-axis at \(A\).</p>
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<p>Find the exact area of the shaded region bounded by the curve and the coordinate axes.<span class="inline-marks">[5]</span></p>

The diagram shows part of the curve \(y=\dfrac{4-x}{\sqrt{x+1}}\), which meets the positive \(x\)-axis at \(A\).

Find the exact area of the shaded region bounded by the curve and the coordinate axes.

Radical curve ending at the x-axis

<p>The diagram shows part of the curve \(y=(x-1)^2(4-x)\).</p>
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<p>Find the area of the shaded region bounded by the curve and the \(x\)-axis between \(A\) and \(B\).<span class="inline-marks">[4]</span></p>

The diagram shows part of the curve \(y=(x-1)^2(4-x)\).

Find the area of the shaded region bounded by the curve and the \(x\)-axis between \(A\) and \(B\).

Repeated-root cubic region

<p>The diagram shows part of the curve \(y=x(x-1)(x-3)\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q28-parts-item-1" data-list-item-path="amarea-q28-parts-item-1"><div class="list-item-paragraph">State the \(x\)-coordinates of the points where the curve meets the \(x\)-axis.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amarea-q28-parts-item-2" data-list-item-path="amarea-q28-parts-item-2"><div class="list-item-paragraph">Find the total area of the two shaded regions.<span class="inline-marks">[6]</span></div></li></ol>

The diagram shows part of the curve \(y=x(x-1)(x-3)\).

State the \(x\)-coordinates of the points where the curve meets the \(x\)-axis.

Find the total area of the two shaded regions.

Total area of two cubic lobes

<p>The diagram shows part of the curve \(y=x^3-6x^2+9x+4\).</p>
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<p>Find the area of the shaded region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x=3\).<span class="inline-marks">[4]</span></p>

The diagram shows part of the curve \(y=x^3-6x^2+9x+4\).

Find the area of the shaded region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x=3\).

Cubic arch between two vertical boundaries

<p>The diagram shows part of the curve \(y=(x+1)(5-x)\).</p>
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<p>Find the area of the shaded region bounded by the curve, the coordinate axes and the line \(x=3\).<span class="inline-marks">[4]</span></p>

The diagram shows part of the curve \(y=(x+1)(5-x)\).

Find the area of the shaded region bounded by the curve, the coordinate axes and the line \(x=3\).

Parabolic region beside the y-axis

<p>The diagram shows part of the curve \(y=\dfrac{20}{(x+2)^2}\).</p>
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<p>Find the area of the shaded region bounded by the curve, the coordinate axes and the line \(x=3\).<span class="inline-marks">[4]</span></p>

The diagram shows part of the curve \(y=\dfrac{20}{(x+2)^2}\).

Inverse-square curve between two vertical boundaries

<p>The diagram shows part of the curve \(y=16-x^4\), which meets the positive \(x\)-axis at \(A\).</p>
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<p>Find the area of the shaded region bounded by the curve and the coordinate axes.<span class="inline-marks">[4]</span></p>

The diagram shows part of the curve \(y=16-x^4\), which meets the positive \(x\)-axis at \(A\).

Find the area of the shaded region bounded by the curve and the coordinate axes.

Quartic region ending at an intercept

<p>The curve \(y=kx-x^2\), where&nbsp;\(k&gt;0\), meets the positive \(x\)-axis at \(A(4,\ 0)\).</p>
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<ol class="i-list"><li data-list-item-id="amarea-q04-parts-item-1" data-list-item-path="amarea-q04-parts-item-1"><div class="list-item-paragraph">Find the value of \(k\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="amarea-q04-parts-item-2" data-list-item-path="amarea-q04-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the \(x\)-axis.<span class="inline-marks">[4]</span></div></li></ol>

The curve \(y=kx-x^2\), where \(k>0\), meets the positive \(x\)-axis at \(A(4,\ 0)\).

Hence find the area enclosed by the curve and the \(x\)-axis.

Parameter from an intercept and the enclosed area

<p>Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.</p>

Find the total area enclosed by the curve \(y = (3x - 2)(x + 2)\) and the \(x\)-axis.

<p>Find the area bounded by the curve \(y = 6x^{2} + 5x + 6\), the \(x\)-axis, the line \(x = -2\) and the line \(x = 2\).</p>

Find the area bounded by the curve \(y = 6x^{2} + 5x + 6\), the \(x\)-axis, the line \(x = -2\) and the line \(x = 2\).

<p>Find the area bounded by the curve \(y = x - 2x^2\) and the positive \(x\)-axis.</p>

Find the area bounded by the curve \(y = x - 2x^2\) and the positive \(x\)-axis.

<p>Find the areas between the following curves and the \(x\)-axis, within the given boundaries.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(y = 25 - x^2\), \(x = -4\), \(x = 1\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(y = (x - 3)(x + 4)(x + 8)\), \(x = -8\), \(x = 3\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(y = x^3 + 2\), \(x = -1\), \(x = 2\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(2y = 2 + x^2\), \(x = -1\), \(x = 1\)</div></li></ol>

Find the areas between the following curves and the \(x\)-axis, within the given boundaries.

\(y = (x - 3)(x + 4)(x + 8)\), \(x = -8\), \(x = 3\)

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the curve \(y = \sqrt{x + 5}\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the area bounded by the curve, the \(x\)-axis and the \(y\)-axis.</div></li></ol>

Find the area bounded by the curve, the \(x\)-axis and the \(y\)-axis.

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the curve \(y = (x + 10)(x - 8)\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the area bounded by the curve and the \(x\)-axis.</div></li></ol>

Sketch the curve \(y = (x + 10)(x - 8)\).

Find the area bounded by the curve and the \(x\)-axis.

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the curve \(y = 9 - x^2\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the area bounded by the curve and the \(x\)-axis.</div></li></ol>

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch the curve \(y = x(x - 3)(x - 5)\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the following integrals.</div><ol class="i-list"><li data-list-item-id="subitem-i" data-list-item-path="item-b/subitem-i"><div class="list-item-paragraph">\(\displaystyle\displaystyle\int_{0}^{3} x^3 - 9x^2 + 18x \, \mathrm{d}x\)</div></li><li data-list-item-id="subitem-ii" data-list-item-path="item-b/subitem-ii"><div class="list-item-paragraph">\(\displaystyle\displaystyle\int_{3}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x\)</div></li><li data-list-item-id="subitem-iii" data-list-item-path="item-b/subitem-iii"><div class="list-item-paragraph">\(\displaystyle\displaystyle\int_{0}^{6} x^3 - 9x^2 + 18x \, \mathrm{d}x\)</div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Explain why the area between the curve and the \(x\)-axis in the interval \(0 \le x \le 6\) is not the answer to <strong>(b)(iii)</strong>. And find the area between the curve and the \(x\)-axis for the interval \(0 \le x \le 6\).</div></li></ol>
<p>In part (c), use the curve \(y=x^3-9x^2+18x=x(x-3)(x-6)\) from part (b). The sketch in part (a) is a separate curve.</p>

Sketch the curve \(y = x(x - 3)(x - 5)\).

Explain why the area between the curve and the \(x\)-axis in the interval \(0 \le x \le 6\) is not the answer to **(b)(iii)**. And find the area between the curve and the \(x\)-axis for the interval \(0 \le x \le 6\).

In part (c), use the curve \(y=x^3-9x^2+18x=x(x-3)(x-6)\) from part (b). The sketch in part (a) is a separate curve.

Understanding Results of Definite Integrals

Definite Integral

Polynomial Functions and Key Features

Differentiation

<p>Find the area between each curve at the \(x\)-axis between the values of \(x\) shown:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(y = 16 - x^2\), \(0 \leq x \leq 4\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(y = \sqrt{x}\), \(4 \leq x \leq 16\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(y = e^{2x}\), \(\ln 2 \leq x \leq \ln 4\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(y = \frac{1}{x^2} + 4\), \(2 \leq x \leq 6\)</div></li></ol>

Find the area between each curve at the \(x\)-axis between the values of \(x\) shown:

\(y = e^{2x}\), \(\ln 2 \leq x \leq \ln 4\)

\(y = \frac{1}{x^2} + 4\), \(2 \leq x \leq 6\)

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