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7. (a) Find \( \int (2x - 1)^{\frac{3}{2}} \, dx \) giving your answer in its simplest form - Edexcel - A-Level Maths Pure - Question 1 - 2016 - Paper 4
Question 1
7. (a) Find \( \int (2x - 1)^{\frac{3}{2}} \, dx \) giving your answer in its simplest form. (b) Find the value of \( k \). (c) Find the exact value of the volume ... show full transcript
Worked Solution & Example Answer:7. (a) Find \( \int (2x - 1)^{\frac{3}{2}} \, dx \) giving your answer in its simplest form - Edexcel - A-Level Maths Pure - Question 1 - 2016 - Paper 4
Step 1
Find \( \int (2x - 1)^{\frac{3}{2}} \, dx \)
Answer
To find the integral, we will use substitution. Let:
[ u = 2x - 1 ] [ du = 2 , dx \Rightarrow dx = \frac{du}{2} ]
The integral becomes:
[ \int (2x - 1)^{\frac{3}{2}} , dx = \frac{1}{2} \int u^{\frac{3}{2}} , du ]
Now we can integrate:
[ \frac{1}{2} \cdot \frac{u^{\frac{5}{2}}}{\frac{5}{2}} = \frac{1}{5} u^{\frac{5}{2}} + C = \frac{1}{5} (2x - 1)^{\frac{5}{2}} + C ]
Step 2
Find the value of \( k \) where the curve cuts the line \( y = 8 \)
Answer
To find ( k ), we set the equation of the curve equal to 8:
[ 8 = (2x - 1)^{\frac{3}{2}} ]
Cubing both sides:
[ 8^2 = (2x - 1)^{3} \Rightarrow 64 = (2x - 1)^{3} ]
Taking the cube root:
[ 2x - 1 = 4 \Rightarrow 2x = 5 \Rightarrow x = \frac{5}{2} ]
Since point P is given as ( P(k, 8) ), we find that ( k = \frac{5}{2} ).
Step 3
Find the exact value of the volume of the solid generated.
Answer
The volume of the solid generated by rotating the region ( S ) around the x-axis can be calculated using the formula:
[ V = \pi \int_{\frac{1}{2}}^{\frac{5}{2}} (y^{2}) , dx ]
Substituting for ( y ):
[ V = \pi \int_{\frac{1}{2}}^{\frac{5}{2}} ((2x - 1)^{\frac{3}{2}})^{2} , dx ]
This simplifies to:
[ V = \pi \int_{\frac{1}{2}}^{\frac{5}{2}} (2x - 1)^{3} , dx ]
Calculating the integral:
[ V = \pi \left[ \frac{(2x - 1)^{4}}{8} \right]_{\frac{1}{2}}^{\frac{5}{2}} ]
Evaluating at the limits:
[ V = \pi \left[ \frac{(4)^{4}}{8} - \frac{(0)^{4}}{8} \right] = \pi \cdot \frac{256}{8} = 32\pi ]