1. Express the given integral as the limit of a Riemann sum but do not evaluate:
integral[0 to 3]((x^3 - 6x)dx)
2.Use the Fundamental Theorem to evaluate integral[0 to 3]((x^3 - 6x)dx).(Your answer must include the antiderivative.)
If there are n rectangles and we use right-side values, then
the width of each rectangle is 3/n
xi=(3/n)*i
so the sum is the limit as n->∞ of
n
∑ f(xi)*∆x = ∑[((3/n)i)^3-6((3/n)i)]*(3/n) = (27/n^4)∑(3i^4-2i^2)
i=1
You can evaluate that sum using the formulas
n
∑ i^2 = (2n^3+3n^2+n)/6
i=1
n
∑ i^4 = (6n^5+15n^4+10n^3-n)/30
i=1
But it's a lot less work to use the integral
∫[0,3] x^3-6x dx = x^4/4-3x^2 [0,3] = 81/4-3*9 = -27/4
1. To express the given integral as the limit of a Riemann sum, we divide the interval [0, 3] into n subintervals of equal width Δx = (3-0)/n. Let xi be the right endpoint of the i-th subinterval. Then, the Riemann sum is given by:
Σ {i=1 to n} f(xi)Δx
In this case, f(x) = x^3 - 6x. Therefore, the Riemann sum is given by:
Σ {i=1 to n} (xi^3 - 6xi)Δx
2. To evaluate the given integral using the Fundamental Theorem of Calculus, we need to find the antiderivative of the integrand f(x) = x^3 - 6x.
Let F(x) be the antiderivative of f(x). Then, we have:
F(x) = ∫ f(x) dx
To find F(x), we can use the power rule for integration. Applying the power rule, we get:
F(x) = (1/4)x^4 - (6/2)x^2 + C
where C is the constant of integration.
Using the Fundamental Theorem of Calculus, we can evaluate the definite integral by subtracting the value of the antiderivative at the upper limit (3) from the value at the lower limit (0):
∫[0 to 3] (x^3 - 6x) dx = F(3) - F(0)
Substituting the antiderivative we found earlier, we have:
F(3) - F(0) = [(1/4)(3^4) - (6/2)(3^2)] - [(1/4)(0^4) - (6/2)(0^2)]
Simplifying further, we get:
F(3) - F(0) = (81/4) - 27
Therefore, the value of the definite integral is:
∫[0 to 3] (x^3 - 6x) dx = (81/4) - 27
1. To express the given integral as the limit of a Riemann sum, we first need to understand that a Riemann sum is a method for approximating the definite integral of a function. It involves dividing the interval of integration into smaller subintervals and approximating the area under the curve of the function within each subinterval using rectangles.
To begin, let's divide the interval [0, 3] into n subintervals of equal width. Each subinterval has a width of Δx = (3-0)/n = 3/n. We can denote the left endpoint of each subinterval as xi, where i ranges from 0 to n-1.
Now, we need to approximate the function value at each xi within their respective subintervals. For this problem, we evaluate the function (x^3 - 6x) at each xi.
So, for the i-th subinterval, the function value is (xi^3 - 6xi).
Next, we multiply the function value by the width of each subinterval to calculate the area of each rectangle. Thus, the area of the i-th rectangle is ((xi^3 - 6xi) * Δx).
Finally, we add up the areas of all n rectangles to get the Riemann sum:
Rn = (Σ((xi^3 - 6xi) * Δx)) for i = 0 to n-1
The limit of this Riemann sum as n approaches infinity will give us the exact value of the integral. However, in this case, we are only asked to express the integral as the limit of a Riemann sum without evaluating it.
2. To evaluate the integral [0 to 3]((x^3 - 6x)dx) using the Fundamental Theorem, we need to find the antiderivative of the function.
First, let's find the antiderivative of x^3. We use the power rule for integration, which states that the antiderivative of x^n (where n is any real number except -1) is (1/(n+1)) * x^(n+1).
So, the antiderivative of x^3 is (1/4) * x^4.
Next, let's find the antiderivative of -6x. The power rule still applies here, but with n = 1.
So, the antiderivative of -6x is (-6/2) * x^2 = -3x^2.
Now that we have the antiderivative, we can evaluate the definite integral using the Fundamental Theorem:
integral[0 to 3]((x^3 - 6x)dx) = F(3) - F(0)
where F(x) is the antiderivative of the function x^3 - 6x.
F(x) = (1/4) * x^4 - 3x^2
Therefore, evaluating the integral gives:
integral[0 to 3]((x^3 - 6x)dx) = F(3) - F(0)
= [(1/4) * 3^4 - 3 * 3^2] - [(1/4) * 0^4 - 3 * 0^2]
= [(1/4) * 81 - 27] - 0
= (81/4 - 108/4)
= -27/4.
Hence, the value of the integral is -27/4, and the antiderivative is (1/4) * x^4 - 3x^2.