Find b such that the line y=b divides the region bounded by the graphs of the two equations into

Find b such that the line y=b divides the region bounded by...

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Key Concepts

Absolute Value Functions and Symmetry

Absolute value functions, such as those expressed as a transformation of |x|, often yield graphs that are symmetric about a vertical axis. This symmetry can simplify the process of calculating areas, as it allows for the evaluation of the integral over half the region and then doubling the result.

Piecewise Functions

When dealing with absolute value functions, it is common to express them as piecewise functions. This piecewise form makes it easier to handle the different behaviors of the function over distinct intervals, allowing for a more straightforward setup of the area integrals.

Definite Integrals

Definite integrals are used to calculate the area under a curve within specific bounds. In problems involving areas bounded by curves, setting up and evaluating the necessary integral is crucial to obtain the total area or portions of it.

Horizontal Partitioning of Regions

Dividing a region into subregions of equal area by a horizontal line involves determining the point (or level) at which a horizontal cut results in two segments with equal area. This requires setting up an equation where the area above or below the horizontal line equals half the total area.

Transcript

00:01 In this question, we have a line y is equal to b that divides a region bounded by the graphs of the equations y is equals to 9 minus mod x and y is equal to 0 into two equal areas.

00:19 We are required to find the value of b.

00:23 So let's see how to solve this question.

00:25 First of all, let's draw the graph for y is equal to 9 minus mod x and y is equal to 0 and the graph is shown below.

00:34 So this is the graph for these two equations.

00:37 This red line represents y is equals to 9 minus mod x and this blue line represents y is equal to 0 and this black line represents y is equals to b.

00:54 Since the region bounded by the curves of y is equal to 9 minus mod x and y is equal to 0 lies between x is equal to minus 9.

01:05 And x is equals to 9 and it is symmetric about y x's.

01:11 Therefore, the expression to calculate area can be written as integration minus 9 to 9, 9 minus x d x.

01:25 So this will be equals to 2 into integration 0 to 9, 9 minus x d x.

01:32 So its integration will be equal to 2 into 9x.

01:36 2 into 9x minus x square upon 2 and the limits are 0 to 9.

01:44 Now substitute the limits.

01:46 So we get 2 into 81 minus 81 upon 2 which will be equals to 81.

01:57 And it is given that the line y equals to b divides this area into two equal parts...