difinite integral f(x)=8-4�ã-3:(X) from x=0 to x=8
To find the definite integral of the function f(x) = 8 - 4√(3x), from x = 0 to x = 8, you need to follow these steps:
1. Start with the given function: f(x) = 8 - 4√(3x).
2. Apply the power rule for integration, which states that the integral of x^n is (1/(n+1))x^(n+1). In this case, the integral of √(3x) with respect to x is (2/3)√(3x)^3/2.
3. Integrate each term separately. The integral of 8 with respect to x is 8x. The integral of -4√(3x) is (-8/3)√(3x)^(3/2).
4. Apply the limits of integration. To find the definite integral from x = 0 to x = 8, substitute these values into the integrated function.
∫[0 to 8] (8 - 4√(3x)) dx = ∫[0 to 8] 8x dx - ∫[0 to 8] (8/3)√(3x)^(3/2) dx
5. Evaluate each term separately using the power rule for integration.
∫[0 to 8] 8x dx = (8/2)x^2 = 4x^2
∫[0 to 8] (8/3)√(3x)^(3/2) dx = (8/3) * (2/3) * √(3x)^(5/2) = (16/9) * √(3x)^(5/2)
6. Apply the limits of integration to each term.
∫[0 to 8] 8x dx = 4(8^2) - 4(0^2) = 256
∫[0 to 8] (8/3)√(3x)^(3/2) dx = (16/9) * √(3(8))^(5/2) - (16/9) * √(3(0))^(5/2) = (16/9) * √(384) - 0
7. Subtract the lower limit from the upper limit.
256 - 0 + (16/9) * √(384)
Therefore, the definite integral of f(x) = 8 - 4√(3x), from x = 0 to x = 8, is 256 + (16/9) * √(384).