(a) Graph f(x)=x(x+2)(x-1) (b) Find the total area between the graph and the x -axis between x=-

(a) Graph f(x)=x(x+2)(x-1) (b) Find the total area between...

Key Concepts

Absolute Value Integration

When a function crosses the x-axis within the region of interest, direct integration might lead to cancellation of areas. To obtain the total area, one must break the integration interval at the zeros and integrate the absolute value of the function over each subinterval so that areas above and below the x-axis are combined as positive contributions.

Net Area versus Total Area

Net area is the result of directly computing the definite integral, which may cancel out volumes above and below the x-axis, yielding a smaller value or even zero. Total area, on the other hand, is found by integrating the absolute value of the function, ensuring that all regions contribute positively, thereby representing the true accumulated area between the graph and the x-axis.

Graphing Functions

Graphing functions involves plotting their key features such as intercepts, turning points, and the overall trend or direction (increasing/decreasing intervals). For polynomial functions, determining the zeros and analyzing intervals between them are crucial steps to accurately sketch the behavior of the graph.

Polynomial Functions

Polynomial functions are expressions involving variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. They are characterized by their degree, roots (or zeros), and end behavior, which together help in understanding the overall shape and behavior of the function.

Definite Integrals

The definite integral of a function over an interval represents the net 'signed' area between the function's graph and the x-axis, as stated by the Fundamental Theorem of Calculus. It involves calculating the accumulation of quantities and gives a positive contribution where the function is above the axis and a negative one where it is below.

Transcript

0:00 Hi there.

00:02 So for part a of this problem, we're asked just to graph the function here, x times x plus 2 times x minus 1.

00:09 So let's just draw a little graph here.

00:16 They give it to us in factored form.

00:18 So from this, we know the zeros.

00:20 In other words, we know that our x intercepts are at 0, negative 2, and positive 1.

00:35 So we have a third degree polynomial goes through these three points on the x -axis and we know the general shape of cubic's there's no negative sign or anything out here so the general shape is something like this using our zeros here and that certainly will be an accurate enough graph for our purposes here so now for part b we are asked to find a total area between the graph and the x -axis from negative two to one so looking at our graph here since this is negative 2 and this is 1 we're asked to find the total area that's this red piece here and also this green piece here okay so to get that first red piece we're going to need definite integral from negative 2 to 0 of our function x minus 1 all right so again this this here corresponds to this red piece here okay now the integral won't be too hard.

01:53 Before we take the anti -derivative, though, we'd probably want to rewrite this.

01:58 So let's go ahead and do that.

02:03 Let's do this part first.

02:06 We can foil that out.

02:08 We'll get x squared.

02:11 We should get plus x minus two.

02:16 That's from expanding these two parentheses.

02:18 And then everything is multiplied by x.

02:20 So really this will be x to the third.

02:23 This will be an x to the second.

02:25 And this will be two.

02:26 X.