find the integration of absolute x from -4 to 2? could anyone show me the way to get 10?
thanks
integrate x from 0 to +4 to get the absolute value when x is negative
add that to the integral from 0 to 2.
first part
4^2/2 - 0^2/2 = 8
second part
2^2/2 - 0^2/2 = 2
8+2 = 10
thanks a lot..
Certainly, I'd be happy to explain how to find the integration of |x| from -4 to 2 and obtain the result of 10.
To evaluate this definite integral, note that the graph of |x| has a "V" shape and intersects the x-axis at x = 0. So we need to split the integral into two separate regions - one for the positive values of x and one for the negative values of x.
For positive values of x (from 0 to 2), the expression |x| is equal to x. Therefore, the integral can be written as:
∫(0 to 2) x dx
Integrating x with respect to x gives (1/2)x^2 evaluated from 0 to 2:
(1/2)(2^2) - (1/2)(0^2) = 2 - 0 = 2
Now, for negative values of x (from -4 to 0), the expression |x| is equal to -x. Therefore, the integral can be written as:
∫(-4 to 0) -x dx
Note that we need to take the opposite sign because x is negative in this range. Integrating -x with respect to x gives (-1/2)x^2 evaluated from -4 to 0:
(-1/2)(0^2) - (-1/2)(-4^2) = 0 - (-1/2)(16) = 0 - (-8) = 8
To find the total value, we sum up the results from both regions:
2 + 8 = 10
So, the integration of |x| from -4 to 2 is indeed equal to 10.