Use the Fundamental Theorem to evaluate the integral from 0 to 3 of the quantity x cubed minus 6 times x, dx.
(Your answer must include the antiderivative.) Use a graph of the function to explain the geometric meaning of the value of the integral.
To evaluate the integral ∫(0 to 3) (x^3 - 6x) dx using the Fundamental Theorem of Calculus, we first need to find the antiderivative of the integrand.
The antiderivative of x^3 - 6x is obtained by applying the power rule of integration. For any term of the form x^n, the antiderivative is (1/(n+1)) * x^(n+1).
So, the antiderivative of x^3 is (1/4) * x^4, and the antiderivative of -6x is -3x^2.
Now, applying the Fundamental Theorem of Calculus:
∫(0 to 3) (x^3 - 6x) dx = [ (1/4) * x^4 - 3 * x^2 ] evaluated from 0 to 3.
Substituting the limits into the antiderivative, we have:
= [ (1/4) * 3^4 - 3 * 3^2 ] - [ (1/4) * 0^4 - 3 * 0^2 ]
= [ (1/4) * 81 - 3 * 9 ] - [ (1/4) * 0 - 3 * 0 ]
= [ 81/4 - 27 ] - [ 0 - 0 ]
= 81/4 - 27
= 81/4 - 108/4
= -27/4
Therefore, the value of the integral ∫(0 to 3) (x^3 - 6x) dx is -27/4.
Now, let's understand the geometric meaning of this value using a graph of the function. By integrating the function, we are finding the signed area under the curve of the graph between x = 0 and x = 3.
For x^3 - 6x, the graph is a polynomial curve. By finding the area between this curve and the x-axis within the limits of 0 and 3, we get -27/4.
The fact that the integral is negative indicates that the area under the curve is below the x-axis, resulting in a net negative area. This represents a region of "loss" or "opposite effect" in the context of the function. In terms of the graph, the negative value of the integral means that the area between the curve and the x-axis is flipped below the x-axis.
So, the geometric interpretation of the value of the integral -27/4 is that it represents the net negative area between the graph of the function x^3 - 6x and the x-axis from x = 0 to x = 3.