If f(x) is odd and the integral from -3 to 7 of f(x) dx is 11, then the integral from 3 to 7 of f(x) dx is __
If f(x) is odd, then f(-x) = -f(x)
So, integral from -3 to 0 is minus the integral from 0 to 3
So, what's left is just integral from 3 to 7 = 11.
To find the value of the integral from 3 to 7 of f(x) dx, you can use the property of odd functions.
An odd function is one in which the function values are symmetric about the origin. Mathematically, for any x, if f(x) is odd, then f(-x) = -f(x).
Since f(x) is odd, we have:
∫[-3 to 7] f(x) dx = 11
To find the integral from 3 to 7, we will consider the symmetry of the function. Since f(x) is odd, the integral from -3 to 7 is equal to the negative of the integral from -7 to 3, because the function values on those intervals are symmetric.
So we have:
∫[-7 to 3] f(x) dx = -∫[-3 to 7] f(x) dx = -11
Now, since we want to find the integral from 3 to 7, we can simply subtract the integral from -7 to 3 from the integral from -3 to 7:
∫[3 to 7] f(x) dx = ∫[-3 to 7] f(x) dx - ∫[-7 to 3] f(x) dx = 11 - (-11) = 22
Therefore, the value of the integral from 3 to 7 of f(x) dx is 22.