Question
Find the area enclosed between the graph of y = k^2 - x^2 and the x-axis, giving your answer in terms of k.
Answers
Reiny
y = -x^2 + k^2 is a downwards parabola with x-intercepts of ±k
area = ∫ (-x^2 + k^2) dx from -k to +k
= 2∫ (-x^2 + k^2) dx from 0 to +k
= 2[ -x^3/3 + k^2x] from 0 to k
= 2(-k^3/3 + k^3 - 0)
= 2(2k^3/3) = (4/3)k^3
check my algebra
area = ∫ (-x^2 + k^2) dx from -k to +k
= 2∫ (-x^2 + k^2) dx from 0 to +k
= 2[ -x^3/3 + k^2x] from 0 to k
= 2(-k^3/3 + k^3 - 0)
= 2(2k^3/3) = (4/3)k^3
check my algebra