Use the FTC 1 to find the area of the region under the graph of the function
f(x)= 8x-x^2 on [0,8].
I got 244/3 but the answer key says 256/3.
∫[0,8] 8x-x^2 dx
= 4x^2 - 1/3 x^3 [0,8]
= 4*8^2 - 1/3 * 8^3
= 256 - 512/3
= (768-512)/3
= 256/3
Maybe you should have shown your work.
To find the area of the region under the graph of a function using the First Fundamental Theorem of Calculus (FTC 1), you need to know the antiderivative of the function and evaluate it at the upper and lower limits of integration.
Given the function f(x) = 8x - x^2, we can find its antiderivative by applying the power rule for integration and the constant rule:
∫f(x) dx = ∫(8x - x^2) dx = 8∫x dx - ∫x^2 dx
Using the power rule, we have:
= 8(1/2)x^2 - (1/3)x^3 + C
Evaluating this antiderivative at the upper and lower limits of integration, [0, 8], we have:
[8(1/2)(8)^2 - (1/3)(8)^3 + C] - [8(1/2)(0)^2 - (1/3)(0)^3 + C]
Simplifying, we get:
= 64 - (64/3)
= 192/3 - 64/3
= 128/3
So according to the FTC 1, the area under the graph of the function f(x) = 8x - x^2 on [0, 8] is 128/3.
It seems there might have been an error in the answer key if it states that the answer is 256/3.