Question
Use the graph of f shown in the figure to evaluate the integral by interpreting it geometrically.
A function in five continuous parts, where each part is a line segment, is graphed on the x y coordinate plane. It enters the top left of the viewing window in the second quadrant. The first part begins from (negative 4, 2), goes up and to the right, and ends at (negative 1, 3). The second part begins at the point (negative 1, 3), goes down and to the right, and ends at the point (0, 2). The third part begins at the point (0, 2), moves horizontally and to the right, 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-axis at (3, 0), and ends at the point (4, negative 2). The fifth part begins at the point (4, negative 2), goes up and to the right, cuts the x-axis at (7, 0), and ends at the point (10, 2).
int_1^2 (f(x)) d x
a.
1
b.
–1
c.
–2
d.
2
A function in five continuous parts, where each part is a line segment, is graphed on the x y coordinate plane. It enters the top left of the viewing window in the second quadrant. The first part begins from (negative 4, 2), goes up and to the right, and ends at (negative 1, 3). The second part begins at the point (negative 1, 3), goes down and to the right, and ends at the point (0, 2). The third part begins at the point (0, 2), moves horizontally and to the right, 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-axis at (3, 0), and ends at the point (4, negative 2). The fifth part begins at the point (4, negative 2), goes up and to the right, cuts the x-axis at (7, 0), and ends at the point (10, 2).
int_1^2 (f(x)) d x
a.
1
b.
–1
c.
–2
d.
2
Answers
GPT 3.5
To evaluate the integral int_1^2 (f(x)) dx geometrically, we need to find the area under the curve of the function f(x) between x = 1 and x = 2.
Looking at the graph of f, the area under the curve between x = 1 and x = 2 is a trapezoid with a base of 1 unit and a height of 3 units.
The area of a trapezoid can be calculated using the formula: A = 0.5 * (b1 + b2) * h, where b1 and b2 are the two bases and h is the height.
In this case, b1 = 2 and b2 = 3, and h = 1. Plugging these values into the formula:
A = 0.5 * (2 + 3) * 1
A = 0.5 * 5 * 1
A = 2.5
Therefore, the integral int_1^2 (f(x)) dx is equal to 2.5.
The closest answer choice is 2, so the correct answer is d. 2.
Looking at the graph of f, the area under the curve between x = 1 and x = 2 is a trapezoid with a base of 1 unit and a height of 3 units.
The area of a trapezoid can be calculated using the formula: A = 0.5 * (b1 + b2) * h, where b1 and b2 are the two bases and h is the height.
In this case, b1 = 2 and b2 = 3, and h = 1. Plugging these values into the formula:
A = 0.5 * (2 + 3) * 1
A = 0.5 * 5 * 1
A = 2.5
Therefore, the integral int_1^2 (f(x)) dx is equal to 2.5.
The closest answer choice is 2, so the correct answer is d. 2.