Question

Find the area lying above the x axis and below the parabolic curve y= 4x -x^2

a.8
b. 8 1/3
c. 10 2/3
d. 16

B or C not sure??

Answer 1

first, we find where the curve touches the x-axis, or simply get the roots,, to get the roots, equate y = 0:
0 = 4x - x^2
0 = x(4-x)
x = 0 and x = 4
this will be our boundary conditions for integration,, we'll use integration since area under the curve:
integral (4x - x^2) = 2x^2 - (1/3)X^3
at x = 0, the integral becomes
2*0^2 - (1/3)*0^3 = 0
at x = 4 the integral becomes
2*(4^2) - (1/3)*4^3
32 - 64/3
96/3 - 64/3
32/3 = 10 2/3
thus integral from 0 to 4 = 10 2/3 - 0 = 10 2/3

hope this helps~ :)