use the definition of the integral to evaluate

Question

In this question, we will use the definition of the integral to evaluate ∫ −1 1​(1−x^2)dx If we partition the interval [−1, 1] into n subintervals, it is difficult to determine precise expressions for the lower and upper sums due to the fact that f (x) = 1 − x2 is increasing on the interval [−1, 0], and decreasing on the interval [0, 1]. In particular, the expressions will change depending on whether we take an even or odd number of subintervals. One way of dealing with this is to instead compute the integral ∫ 0 1​(1−x^2)dx and exploit the symmetry of the given function f.

(a) Using appropriate summation formulas from lectures, determine the expressions for Ln and Un. Then find the limit values of Ln and Un, and hence determine ∫ 0 1​(1−x^2)dx.

(b) Using your answer in part (a) and the fact that f is an even function (i.e. for any x, f (x) = f (−x)), determine ∫ −1 1​(1−x^2)dx

Answer 1

integral of (1-x^2)dx

=integral of 1 minus integral of x^2

integral of a sum = sum of the integrals

= x -x^3/3 + c

evaluated from -1 to 1

= evaluated from 0 to 1 plus evaluated from -1 to 0

=(1- 1/3) -0 + 0 - (-1+1/3)

= 2/3 + 2/3

=4/3

as it's an even function, just calculate the integral evaluated from 0 to 1, then multiply by 2

= 2/3 times 2 = 4/3

you should get ever closer to 4/3 as n gets ever larger, by calculating areas under the curve

Answer 2

Reference video for detail lesson