Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line $x = 2$. The region $S$ is rotated $360^ ext{o}$ about the x-axis. Use integration to find the exact value of the volume of the solid generated, giving your answer in the form $k \, ext{ln} \, a$, where $k$ and $a$ are constants.

A-Level Edexcel Maths Pure Differentiation: Figure 1 shows the curve with eq

Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line $x = 2$ - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 8

Question 5

$Figure-1-shows-the-curve-with-equation--$$y-=--rac{2x}{-oot{3x^2-+-4}}$$,--$x->-0$--The-finite-region-$S$,-shown-shaded-in-Figure-1,-is-bounded-by-the-curve,-the-x-axis-and-the-line-$x-=-2$-Edexcel-A-Level Maths Pure-Question 5-2012-Paper 8.png$

Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-a... show full transcript

Worked Solution & Example Answer:Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line $x = 2$ - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 8

Step 1

Use of the volume formula

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Answer

To find the volume of the solid generated by rotating the region $S$ about the x-axis, we can use the formula:

$V = rac{1}{3} imes ext{Area} imes ext{Height}.$

So, we set up the integral based on the function:

oot{3x^2 + 4}} \, dx $$.

Step 2

Working with the integral

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Answer

Apply substitutions and calculate the integral. Use:

$V = rac{1}{3} \int_{0}^{2} \frac{2}{3} rac{2x}{(3x^2 + 4)^{1/2}} \, dx$ .

Integrate the above function, which may involve parts or specific substitutions.

Step 3

Final calculation of volume

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Answer

After integrating and substituting the limits, your result should yield:

$V = \frac{1}{3} \left[ \ln(16) - \ln(4) \right] = \frac{1}{3} \ln(4)$ .

Finally, express the answer in the form required:

$V = \frac{2}{3} \pi \ln(4).$

Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line $x = 2$ - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 8

Worked Solution & Example Answer:Figure 1 shows the curve with equation $$y = rac{2x}{ oot{3x^2 + 4}}$$, $x > 0$ The finite region $S$, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line $x = 2$ - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 8

Use of the volume formula

Working with the integral

Final calculation of volume

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