Explanation
1. Find the Points of Intersection:
To determine the limits of integration, we set the two equations equal to each other:
x3=4x2
x3−4x2=0
x2(x−4)=0
The curves intersect at x=0 and x=4.
2. Determine the Upper and Lower Curve:
In the interval [0,4], let's check which curve is above the other by testing a point, e.g., x=1:
3. Set up the Integral:
The area A is given by the integral of the upper curve minus the lower curve:
A=∫04(4x2−x3)dx
4. Evaluate the Integral:
Applying the power rule for integration:
A=[4x33−x44]04
Substitute the upper limit x=4:
A=(4(4)33−(4)44)−(0)
A=4(64)3−2564
A=2563−64
5. Final Calculation:
Find a common denominator to subtract:
A=256−1923
A=643
Correct Option: (b) 64/3
