Show that the area under the curve y=f(x) for 0<=x<=a is twice the area between the curve y=f(x) and the line ay=f(a)x between the points(0,0) and (a,f(a)).
2006-08-28 06:19:23 · 4 answers · asked by dhufiweioq j 1 in Science & Mathematics ➔ Mathematics
It's not. Suppose y = f(x) = x, and suppose a = 6.
The area under the curve y = f(x) for the interval (0,6) is
A = int[f(x) dx] = int(x dx) = (1/2) x^2 = 36/2 = 18
The line ay = f(a)x is 6y = 6x (since f(a) = 6), or y = x.
The area under that line between (0,0) and (6,6) is (1/2)(6^2) = 18.
In this counterexample, the curve y = f(x) and the line ay = f(a)x
are the same, so the difference between the two is zero.
2006-08-28 07:01:27 · answer #1 · answered by bpiguy 7 · 1⤊ 0⤋
Is f(x)=y just some curved function which is unrelated to your linear function?
kindly clarify ay=f(a)x? did you mean f(x)=ay? ie: f(x) is a straight line.
2006-08-28 13:59:38 · answer #2 · answered by Curious 2 · 0⤊ 0⤋
f(a) does not necessarily equal "a". it could be that f(a) = 7a + 3
2006-08-31 13:45:08 · answer #3 · answered by leobloom 1 · 0⤊ 0⤋
what differential does it make?
2006-08-28 13:21:32 · answer #4 · answered by gnyla 2 · 0⤊ 0⤋