Question

use the graph of ( f ) shown in the figure to evaluate the integral by interpreting it geometrically: ( int_{4}^{1} f(x) d x ).
(description: a function in five continuous parts, where each part is a line segment, is graphed on the ( x y ) coard plane. it enters the top left of the viewing window in the second quadrant. the first part begins from (negative 4 goes up and to the right, and ends at (negative 1, 3). the second part begins at the point (negative 1, 3), goes to the right, and ends at the point ( (0,2) ). the third part begins at the point ( (0,2) ), moves horizontally and to the and ends at the point ( (2,2) ). the fourth part begins at the point ( (2,2) ), goes down and to the right, cuts the ( x )-ax ( 0 ), and ends at the point ( (4 ), negative 2). the fifth part begins at the point ( (4 ), negative 2), goes up and to the r the ( x )-axis at ( (7,0) ), and ends at the point ( (10,2) ).)

Explanation:

Step1: Analyze the graph for the interval \([4,10]\)

The integral \(\int_{4}^{10}f(x)dx\) can be interpreted as the net - area between the graph of \(y = f(x)\) and the \(x\) - axis from \(x = 4\) to \(x=10\).
The region from \(x = 4\) to \(x = 7\) is a triangle with base \(b=7 - 4=3\) and height \(h = 2\). The area of a triangle is \(A=\frac{1}{2}bh\). So, \(A_1=\frac{1}{2}(3)(2)=3\).
The region from \(x = 7\) to \(x = 10\) is a triangle with base \(b = 10 - 7=3\) and height \(h=2\). Using the area formula for a triangle \(A=\frac{1}{2}bh\), we get \(A_2=\frac{1}{2}(3)(2)=3\).

Step2: Calculate the integral

Since the integral \(\int_{4}^{10}f(x)dx\) is the sum of the areas of these two non - negative regions (because the function is above the \(x\) - axis for \(x\in[7,10]\) and the area below the \(x\) - axis for \(x\in[4,7]\) is considered as a positive value when calculating the integral in the geometric interpretation), \(\int_{4}^{10}f(x)dx=A_1 + A_2\).
\(\int_{4}^{10}f(x)dx=3+3=6\)

Answer:

6