Definite Integral?

Question

What is the definite integral for:

(2-x^2)dx for the intervals [2,-1]

Answer 1

( integral ) ( 2 - x^2 ) dx

(integral ) 2 dx - ( integral ) x^2 dx

2x (evaluated from 2 to -1) - 1/3 x^3 ( evaluated from 2 to -1 )

2 ( -1 - 2 ) - 1/3 ( ( -1 ) ^3 - ( 2 )^3 )

Hopefully I got the limits in the right order there :)

Answer 2

Before we attempt to find out the definite integral, we must first find out the indefinite integral (integrals that do not require intervals to solve; the indefinite integral of a function is another function whose derivative is the first function). Three rules of integration that will help:

1. For any constant a dx, the indefinite integral (also known as the antiderivative) is ax + C, where C is any non-variable. Why C is included: ax is an antiderivative of a, since the derivative of ax is a. But ax + 4 is also an antiderivative, since the derivative of 4 is 0. We can use 4 or any other number like 5 or 6, and we can generalize this with C - "C" for constant. Note that in definite integrals C is always 0.

2. The sum rule: If we have the sum of n functions to integrate (where n is any integer greater than 1), then we can split the integral into n integrals, each containing the separate integrals.

3. The power rule: If we have x^a, where a is any rational number ("rational number" means integer or fractional number) other than –1, then the integral of (x^a) dx is ((x^(r + 1))/(r + 1)) + C.

The interval that you provided (to determine the definite integral we must first have an interval of integration and the indefinite integral with C = 0) is [2, –1]. The upper limit is less than the lower limit, so we must first reverse the limits and multiply the definite integral obtained by –1 to obtain the final result.

Using these rules and the knowledge that C is always 0 for any definite integral, we can determine that the correct antiderivative to use is 2x – (x^3)/3. For –1, the antiderivative becomes –2 – (–1/3) = –2 + 1/3 = –5/3. For 2, the antiderivative becomes 4 – 8/3 = 4/3. In your case the upper limit of the interval is 2, and the lower limit is –1.

To obtain the definite integral we must subtract the value of the antiderivative for the lower limit from the value of the antiderivative for the upper limit. And 4/3 – (–5/3) = 4/3 + 5/3 = 9/3 = 3, and multiplying by –1 we have –3, the final result.