You are given the four points in the plane A=(−1,−8), B=(4,5),C=(8,−2), and D=(12,2). The graph of the function f(x)f(x) consists of the three line segments AB, BC and CD. Find the integral ∫12−1f(x)dx by interpreting the integral in terms of sums and/or differences of areas of elementary figures.
Sorry for the confusion on the last problem
No problem, I'm here to help! So, we need to find the integral of f(x) from x = -1 to x = 12. To interpret the integral in terms of sums and/or differences of areas of elementary figures, we can break down the interval into smaller sections and calculate the areas individually.
Let's start by breaking down the interval [-1, 12] into three smaller intervals: [-1, 4], [4, 8], and [8, 12].
Now, we want to find the area under the curve f(x) within each of these smaller intervals.
In the interval [-1, 4], the area under f(x) can be calculated as the area of a trapezoid. The base of the trapezoid is (4 - (-1)) = 5 units, and the heights at the endpoints are f(-1) = -8 and f(4) = 5. So, the area of this trapezoid is (5/2) * (|f(-1)| + |f(4)|).
In the interval [4, 8], the area under f(x) is simply the area of a rectangle. The base of the rectangle is (8 - 4) = 4 units, and the height is f(4) = 5. The area of this rectangle is 4 * |f(4)|.
In the interval [8, 12], we can again calculate the area as the area of a trapezoid. The base of the trapezoid is (12 - 8) = 4 units, and the heights at the endpoints are f(8) = -2 and f(12) = 2. So, the area of this trapezoid is (4/2) * (|f(8)| + |f(12)|).
To find the total area, we sum up the areas of all these elementary figures:
Area = (5/2) * (|f(-1)| + |f(4)|) + 4 * |f(4)| + (4/2) * (|f(8)| + |f(12)|)
And that's how we interpret the integral in terms of sums and/or differences of areas of elementary figures.
To find the integral ∫12−1f(x)dx using the given graph of function f(x), we need to interpret the integral in terms of sums and/or differences of areas of elementary figures.
Let's break down the problem step by step:
Step 1: Find the equations of the three line segments AB, BC, and CD.
The equation of a line passing through two points (x1, y1) and (x2, y2) can be found using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
For line AB:
Point A = (-1, -8) and Point B = (4, 5)
Let's find the slope (m) first:
m = (y2 - y1) / (x2 - x1) = (5 - (-8)) / (4 - (-1)) = 13 / 5
Now, let's find the y-intercept (b) using one of the points (A or B):
Using point B (4, 5):
5 = (13/5)(4) + b
b = 5 - (13/5)(4) = -27/5
So, the equation of line AB is: y = (13/5)x - 27/5
Similarly, we can find the equations of line BC and CD:
For line BC: y = (-7/4)x + 23/4
For line CD: y = (1/2)x - 7/2
Step 2: Find the points of intersection of these line segments.
The only point of intersection of these line segments lies on line BC.
To find the x-coordinate of this point, we need to solve the equations of line BC and line AB.
(-7/4)x + 23/4 = (13/5)x - 27/5
Multiplying both sides by 20:
-35x + 115 = 52x - 108
87x = 223
x = 223/87
Now, substitute this value of x into the equation of line BC to find the y-coordinate:
y = (-7/4)(223/87) + 23/4
y = -223/87 + 561/87
y = 338/87
So, the point of intersection of line segments BC and AB is (223/87, 338/87).
Step 3: Determine the areas of the elementary figures.
We need to find the areas of the trapezoids formed by the line segments AB, BC, and CD.
The area of a trapezoid can be calculated using the formula: A = (b1 + b2) * h / 2, where b1 and b2 are the lengths of the parallel sides, and h is the height.
For trapezoid ABDE:
Length of parallel sides: AB and DE
AB = (4 - (-1)) = 5
DE = (12 - 8) = 4
Height: The y-coordinate difference between A and D = (2 - (-8)) = 10
Area ABDE = (5 + 4) * 10 / 2 = 45
For trapezoid BCEF:
Length of parallel sides: BC and EF
BC = (8 - 4) = 4
EF = (12 - 8) = 4
Height: The y-coordinate difference between C and E = (2 - (-2)) = 4
Area BCEF = (4 + 4) * 4 / 2 = 16
Now, we have the areas of the two trapezoids formed by the line segments AB, BC, and CD.
Step 4: Calculate the integral by summing the areas.
The integral of f(x) from x = -1 to x = 12 is equal to the sum of the areas of the trapezoids ABDE and BCEF.
∫12−1f(x)dx = Area ABDE + Area BCEF = 45 + 16 = 61
Therefore, the value of the integral ∫12−1f(x)dx is 61.