Consider the given diagram with
marked region I, II and III. Mark
the correct options
y
y=x?
MI
1 2
Clear Response
Area of region I is 1/3 sq. units
Area of region 1 is 4/3 sq. units
Area of region II is 8/3 sq. units
Area of region III is 16/3 sq. units
Answers
Answer:
Determine the area of a region between two curves by integrating with respect to the independent variable.
Find the area of a compound region.
Determine the area of a region between two curves by integrating with respect to the dependent variable.In Introduction to Integration, we developed the concept of the definite integral to calculate the area below a curve on a given interval. In this section, we expand that idea to calculate the area of more complex regions. We start by finding the area between two curves that are functions of x, beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of y.Area of a Region between Two Curves
Let f(x) and g(x) be continuous functions over an interval \left[a,b\right] such that f(x)\ge g(x) on \left[a,b\right]. We want to find the area between the graphs of the functions, as shown in the following figure.
This figure is a graph in the first quadrant. There are two curves on the graph. The higher curve is labeled “f(x)” and the lower curve is labeled “g(x)”. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b.
Figure 1. The area between the graphs of two functions, f(x) and g(x), on the interval \left[a,b\right].
As we did before, we are going to partition the interval on the x\text{-axis} and approximate the area between the graphs of the functions with rectangles. So, for i=0,1,2\text{,…},n, let P=\left\{{x}_{i}\right\} be a regular partition of \left[a,b\right]. Then, for i=1,2\text{,…},n, choose a point {x}_{i}^{*}\in \left[{x}_{i-1},{x}_{i}\right], and on each interval \left[{x}_{i-1},{x}_{i}\right] construct a rectangle that extends vertically from g({x}_{i}^{*}) to f({x}_{i}^{*}). (Figure)(a) shows the rectangles when {x}_{i}^{*} is selected to be the left endpoint of the interval and n=10. (Figure)(b) shows a representative rectangle in detail.