Question

Use the (limit) definition of the definite integral to evaluate ∫ 2

4
(x 2
−x)dx. You may find some of the following useful: ∑ k=1
n
c=cn,∑ k=1
n
k= 2
n(n+1)
,∑ k=1
n
k 2
= 6
n(n+1)(2n+1)
,∑ k=1
n
k 3
= 4
n 2
(n+1) 2

Answer 1

To solve the integral ∫ from 2 to 4 (x^2 - x) dx, we have to integrate the function x^2 - x with respect to x, from the lower limit 2 to the upper limit 4.

Step 1: Antiderivative
The antiderivative of a function is found by reversing the derivative. The antiderivative of x^2 - x is the function F(x) = (1/3)x^3 - (1/2)x^2. This is found by using the power rule of integration, which states that the integral of x^n dx is (1/n+1)x^(n+1).

Step 2: Evaluating the Definite Integral
The definite integral of a function from a to b is found by evaluating the antiderivative at the upper limit and the lower limit, and subtracting these two results. So, we substitute the limits of integration into the antiderivative:

F(4) = (1/3)(4)^3 - (1/2)(4)^2 = 64/3 - 8 = 64/3 - 24/3 = 40/3

F(2) = (1/3)(2)^3 - (1/2)(2)^2 = 8/3 - 2 = 8/3 - 6/3 = 2/3

Then, the definite integral is F(b) - F(a) = 40/3 - 2/3 = 38/3.

Hence, the integral ∫ from 2 to 4 (x^2 - x) dx is 38/3.