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IB AA HL Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The line \(x=1\) meets the curve \(x=\dfrac{(y-2)^2}{4}\) at \(P\) and \(Q\).</p>
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<p>Find the area enclosed by the curve and the line.<span class="inline-marks">[5]</span></p>

The line \(x=1\) meets the curve \(x=\dfrac{(y-2)^2}{4}\) at \(P\) and \(Q\).

Find the area enclosed by the curve and the line.

Area Bounded by a Curve and Straight Lines

Area Enclosed by the Curve and the Y-Axis

Sideways parabola cut by a vertical line

Integration

<p>The curve \(x=(y-2)^2\) intersects the line \(x=4-y\) at the points \(P\) and \(Q\).</p>
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data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="amarea-q25-parts-item-1" data-list-item-path="amarea-q25-parts-item-1"><div class="list-item-paragraph">Find the coordinates of \(P\) and \(Q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amarea-q25-parts-item-2" data-list-item-path="amarea-q25-parts-item-2"><div class="list-item-paragraph">Hence find the area enclosed by the curve and the line.<span class="inline-marks">[5]</span></div></li></ol>

The curve \(x=(y-2)^2\) intersects the line \(x=4-y\) at the points \(P\) and \(Q\).

Hence find the area enclosed by the curve and the line.

Intersection of a Line and a Curve

Sideways parabola and an oblique line

Quadratic Functions, Equations and Inequalities

<p>The curve \((y-4)^2=3x\) touches the \(y\)-axis at \(P(0,\ 4)\). The line \(y=1\) meets the curve at \(Q\).</p>
<div class="graph-block" 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data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Find the area of the shaded region bounded by the curve, the \(y\)-axis and the line \(y=1\).<span class="inline-marks">[4]</span></p>

The curve \((y-4)^2=3x\) touches the \(y\)-axis at \(P(0,\ 4)\). The line \(y=1\) meets the curve at \(Q\).

Find the area of the shaded region bounded by the curve, the \(y\)-axis and the line \(y=1\).

Sideways parabola beside the y-axis

<p>The diagram shows part of the curve \(x=y^2-3y\) and the line \(x+y=3\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768008012336-hpxi3pcmgut.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMDgwMTIzMzYtaHB4aTNwY21ndXQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTUzNSwiZXhwIjoxNzkyNzMxODM1fQ.VoDPxrzllB9bXl_oCCOWSksinRDexVdSY5flM7L7m9E" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>If the line and the curve intersect at \(P\) and \(Q\), find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">coordinates of \(P\) and \(Q\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">area of the shaded region.</span></div></li></ol>

The diagram shows part of the curve \(x=y^2-3y\) and the line \(x+y=3\).

If the line and the curve intersect at \(P\) and \(Q\), find the

<p>The line \(y = \frac{1}{2}x\) is the tangent to the curve \(y^2 = x - 1\) at the point \((2, 1)\). Calculate the area enclosed by the curve, this tangent and the \(x\)-axis.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767948563680-h83o3dq0u9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDg1NjM2ODAtaDgzbzNkcTB1OS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzMxNTM1LCJleHAiOjE3OTI3MzE4MzV9.cH9L6mCgz4dCE8M9peto6NpRlfAymMJPc-d8yH323TA" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The line \(y = \frac{1}{2}x\) is the tangent to the curve \(y^2 = x - 1\) at the point \((2, 1)\). Calculate the area enclosed by the curve, this tangent and the \(x\)-axis.

<p>The figure shows the two curves \(y^2 = 4x\) and \(2x + y = 4\). Find the area of the region bounded by the curves.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767948479707-h3890jvodgg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDg0Nzk3MDctaDM4OTBqdm9kZ2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTUzNSwiZXhwIjoxNzkyNzMxODM1fQ.OWxbgwqk7U57TW7Kc157Hpm0qSTkYf1Ws5oVLtxsHPE" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the two curves \(y^2 = 4x\) and \(2x + y = 4\). Find the area of the region bounded by the curves.

<p>The figure shows the curve \(x = e^{y} - y - 3\) which intersects the \(y\)-axis at \(1.505\) and \(-2.948\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = 2\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767947879166-t5amt72bst.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc4NzkxNjYtdDVhbXQ3MmJzdC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzMxNTM1LCJleHAiOjE3OTI3MzE4MzV9.oNKt8Cc-5nb_d0rYaxkJTFirQcBKiiQv879sQj0WL5g" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the curve \(x = e^{y} - y - 3\) which intersects the \(y\)-axis at \(1.505\) and \(-2.948\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = 2\).

<p>The figure shows the curve \(x = 32 - 8y^2\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = -4\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767947762609-s10h0b1ppea.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc3NjI2MDktczEwaDBiMXBwZWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTUzNSwiZXhwIjoxNzkyNzMxODM1fQ.92iUdT_j-O70qQ3dbY1gvfeyJnb8XOO4jx008bJHROY" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the curve \(x = 32 - 8y^2\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = -4\).

<p>The figure shows the curve \(x = y^2 - 9\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = 4\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767947608878-t8whptp2kul.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc2MDg4NzgtdDh3aHB0cDJrdWwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTUzNSwiZXhwIjoxNzkyNzMxODM1fQ.v4zwk6WNSQe6Xh4xmfnLTpML5zmnyKt6gnhV5I9M9Os" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The figure shows the curve \(x = y^2 - 9\). Find the area of the region bounded by the curve, \(y\)-axis and the line \(y = 4\).

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