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4. (a) Express $$\frac{25}{x^2(2x + 1)}$$ in partial fractions - Edexcel - A-Level Maths: Pure - Question 7 - 2014 - Paper 8
Question 7
4. (a) Express $$\frac{25}{x^2(2x + 1)}$$ in partial fractions. (b) Use calculus to find the exact volume of the solid of revolution generated, giving your ans... show full transcript
Worked Solution & Example Answer:4. (a) Express $$\frac{25}{x^2(2x + 1)}$$ in partial fractions - Edexcel - A-Level Maths: Pure - Question 7 - 2014 - Paper 8
Step 1
Express \(\frac{25}{x^2(2x + 1)}\) in partial fractions.
Answer
To express the given fraction in partial fractions, we can assume it takes the following form:
Next, we multiply through by the denominator, resulting in:
Expanding both sides gives:
To find the coefficients, we need to collect terms:
Now, equate coefficients from both sides:
- For the constant term:
- For the coefficient of (x):
- For the coefficient of (x^2):
Thus, the partial fraction decomposition is:
Step 2
Use calculus to find the exact volume of the solid of revolution generated.
Answer
The volume (V) of the solid generated by rotating region (R) around the x-axis can be computed using the formula:
Here, the function defined by the curve (C) is:
Thus, we have:
This simplifies to:
Computing the integral, we can separate the terms:
Using the substitution (u = 2x + 1) yields:
Changing the limits accordingly, when (x=1), (u=3) and when (x=4), (u=9). Thus, the entry becomes:
Evaluating the integral gives:
Finally, we can express the volume in the required form:
where:
a = 0, b = \frac{75\pi}{2}, c = 3.