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The curve C has equation $y = 2x^2$ for $0 \\leq x \\leq 4$ - CIE - A-Level Further Maths - Question 2 - 2012 - Paper 1

Question 2

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The curve C has equation $y = 2x^2$ for $0 \\leq x \\leq 4$. Find (i) the mean value of $y$ with respect to $x$ for $0 \\leq x \\leq 4$. (ii) the $y$-coordinate of... show full transcript

Worked Solution & Example Answer:The curve C has equation $y = 2x^2$ for $0 \\leq x \\leq 4$ - CIE - A-Level Further Maths - Question 2 - 2012 - Paper 1

Step 1

the mean value of $y$ with respect to $x$ for $0 \leq x \leq 4$

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Answer

To find the mean value of $y$ over the interval from $0$ to $4$ , we use the formula:

$\text{Mean Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx$

Here, $f(x) = 2x^2$ , $a = 0$ , and $b = 4$ . Thus, we compute:

$\text{Mean Value} = \frac{1}{4-0} \int_{0}^{4} 2x^2 \, dx$

Calculating the integral:

Compute the antiderivative: $\int 2x^2 \, dx = \frac{2}{3} x^3 + C$
Evaluate from $0$ to $4$ : $= \left[ \frac{2}{3} (4)^3 - \frac{2}{3} (0)^3 \right] = \left[ \frac{2}{3} \cdot 64 \right] = \frac{128}{3}$

Therefore,

$\text{Mean Value} = \frac{1}{4} \cdot \frac{128}{3} = \frac{32}{3}.$

Step 2

the $y$-coordinate of the centroid of the region enclosed by C, the line $x = 4$ and the $x$-axis

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Answer

The $y$ -coordinate of the centroid is given by the formula:

$\bar{y} = \frac{1}{A} \int_{a}^{b} y \, dx$

Where $A$ is the area under the curve from $x = 0$ to $x = 4$ .

First, we find the area $A$ : $A = \int_{0}^{4} 2x^2 \, dx = \left[ \frac{2}{3} x^3 \right]_{0}^{4} = \frac{128}{3}.$
Then we compute the value of $ar{y}$ : $\bar{y} = \frac{1}{A} \int_{0}^{4} 2x^2 \, dx = \frac{1}{\frac{128}{3}} \cdot \frac{128}{3} = \frac{1}{2} \cdot \int_{0}^{4} 2x^2 \, dx = \frac{1}{2} \cdot \frac{128}{3} = \frac{64}{3}.$

Thus, the $y$ -coordinate of the centroid is: $\bar{y} = \frac{16 \cdot 3}{32} = \frac{3}{2}.$

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The curve C has equation $y = 2x^2$ for $0 \\leq x \\leq 4$ - CIE - A-Level Further Maths - Question 2 - 2012 - Paper 1

Worked Solution & Example Answer:The curve C has equation $y = 2x^2$ for $0 \\leq x \\leq 4$ - CIE - A-Level Further Maths - Question 2 - 2012 - Paper 1

the mean value of $y$ with respect to $x$ for $0 \leq x \leq 4$

the $y$-coordinate of the centroid of the region enclosed by C, the line $x = 4$ and the $x$-axis

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