A-Level Edexcel Maths Pure Circles: Figure 1 shows a sketch of part

Figure 1 shows a sketch of part of the curve C with equation $y = x(x - 1)(x - 5)$ - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 2

Question 9

Figure 1 shows a sketch of part of the curve C with equation $y = x(x - 1)(x - 5)$. Use calculus to find the total area of the finite region, shown shaded in Figure... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of part of the curve C with equation $y = x(x - 1)(x - 5)$ - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 2

Step 1

Step 1: Set Up the Integral

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Answer

To find the area between the curve and the x-axis from $x = 0$ to $x = 2$ , we need to evaluate the definite integral:

$A = \int_0^2 y \, dx = \int_0^2 x(x - 1)(x - 5) \, dx$

First, we will expand the function:

$y = x(x - 1)(x - 5) = x(x^2 - 6x + 5) = x^3 - 6x^2 + 5x$ . Hence, the integral becomes:

$A = \int_0^2 (x^3 - 6x^2 + 5x) \, dx$

Step 2

Step 2: Calculate the Integral

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Answer

Now we will compute the integral:

$A = \int (x^3) \, dx - 6 \int (x^2) \, dx + 5 \int (x) \, dx$

Evaluating each term:

$\int x^3 \, dx = \frac{x^4}{4}$ ,
$-6 \int x^2 \, dx = -6 \cdot \frac{x^3}{3} = -2x^3$ ,
$5 \int x \, dx = \frac{5x^2}{2}$ .

Combining, we have:

$A = \left[ \frac{x^4}{4} - 2x^3 + \frac{5x^2}{2} \right]_0^2$

Step 3

Step 3: Evaluate the Definite Integral

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Answer

Now substituting the limits of integration:

$A = \left( \frac{2^4}{4} - 2(2^3) + \frac{5(2^2)}{2} \right) - \left( 0 \right)$

Calculating each term:

$\frac{2^4}{4} = 4$ ,
$-2(8) = -16$ ,
$\frac{5(4)}{2} = 10$ .

Combining these values:

$A = 4 - 16 + 10 = -2.$

However, since we need the area, we take the absolute value:

Thus, the area is $2$ .

Step 4

Step 4: Conclusion

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Answer

The total area of the finite region between the curve C, the x-axis, and the vertical lines at $x = 0$ and $x = 2$ is:

$$ \text{Total Area} = 2 $.

Figure 1 shows a sketch of part of the curve C with equation $y = x(x - 1)(x - 5)$ - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a sketch of part of the curve C with equation $y = x(x - 1)(x - 5)$ - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 2

Step 1: Set Up the Integral

Step 2: Calculate the Integral

Step 3: Evaluate the Definite Integral

Step 4: Conclusion

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