using geometric formula evaluate 4-x and 2x-8 when integral is from x=0 to 6
not sure what you mean by
evaluate 4-x and 2x-8
Maybe a look at the graphs will help clarify your thinking:
http://www.wolframalpha.com/input/?i=plot+y%3D4-x%2C+y%3D2x-8
To evaluate the expressions 4-x and 2x-8 using geometric formulas, we need to calculate the definite integrals of these expressions from x=0 to x=6.
Let's start with the expression 4-x:
∫(4-x) dx
To integrate this expression, we need to use the power rule of integration. According to the power rule, ∫x^n dx = (1/n+1) x^(n+1) + C, where C is the constant of integration.
So, using the power rule, the integral of (4-x) dx becomes:
∫(4-x) dx = 4x - (1/2)x^2 + C1
Next, let's evaluate the expression 2x-8:
∫(2x-8) dx
Again, we need to apply the power rule for integration. Using the power rule, the integral of (2x-8) dx becomes:
∫(2x-8) dx = x^2 - 8x + C2
Now, we can evaluate both of these expressions within the given range of integration, from x=0 to x=6.
For the expression 4-x:
∫(4-x) dx = [4x - (1/2)x^2] evaluated from 0 to 6
Plugging in the upper limit (6):
[4(6) - (1/2)(6)^2] = 24 - 18 = 6
Plugging in the lower limit (0):
[4(0) - (1/2)(0)^2] = 0
Thus, the value of ∫(4-x) dx from x=0 to x=6 is 6.
For the expression 2x-8:
∫(2x-8) dx = [x^2 - 8x] evaluated from 0 to 6
Plugging in the upper limit (6):
[(6)^2 - 8(6)] = 36 - 48 = -12
Plugging in the lower limit (0):
[(0)^2 - 8(0)] = 0
Thus, the value of ∫(2x-8) dx from x=0 to x=6 is -12.
In conclusion, by using geometric formulas and evaluating the definite integrals, we found that the values of the expressions 4-x and 2x-8, within the range of x=0 to x=6, are 6 and -12, respectively.