the graph of a piecewise linear function f, for -1<x<4. what is the value of integral (-1 to 4) F(x) dx?
the interval i gave is grerater than or equal to and less than and equal to. i didn't know how to put the underscore under the symbol. for the integral, -1 is on the bottom of the integral symbol and 4 is on the top.
anyway, since i can't draw the graph, here are the points: (-1,0) (0,2) (1,2) (2,0) (3,-1) (4,-1)
the ilnes connecting the dots is straight, no curves and stuff.
if you look at the overall graph, i can separate it into triangles and squares. so i did that and i added up the total area of each shape and got
5.5.
the second way i did it:
when i drew the graph, part of it is under the x-axis so when i was trying to find the area, i subtracted that from the the area of the other part of the graph and i got 4.
this is a multiple choice question, both the answers were in there, so now i'm confused. which one is right?
sorry if my explanation is to wordy or confusion. did the best i can
If I'm not mistaken, wouldn't the answer be 2.5? I have the exact same problem in front of me; I approximated using rectangles (half values for triangles). The area above the curve was 4, and below was 1.5... shouldn't you subtract the 1.5 from the 4?
my name ! :)
Well, aren't integrals just fancy ways of adding up areas? So, in this case, we're trying to find the area under the graph of the function f from -1 to 4.
You've got two different calculations here, but the key is to ensure consistency in the method you use. If you're subtracting the area below the x-axis, you'll need to subtract that from the area of the other part of the graph as well.
Considering your first calculation, where you added up the total area of each shape, you arrived at 5.5. That's the total area under the graph, including both the positive and negative parts. So, if you want to find the area solely between the x-axis and the graph, you'll need to subtract the area below the x-axis.
In your second calculation, you subtracted the area below the x-axis from the area above it and got 4. That's only considering the positive part of the graph. So, it seems like this approach misses the negative part of the graph.
In conclusion, based on your description, the correct answer should be 5.5 as that includes both the positive and negative parts of the graph.
To find the value of the integral of the piecewise linear function f(x) from -1 to 4, we need to calculate the total area under the curve.
Based on the given points, we can see that the graph consists of two parts: a triangle and a rectangle.
First, let's calculate the area of the triangle. The base of the triangle is 2 units (from x = 0 to x = 2) and the height is 2 units. So the area of the triangle is (1/2) * 2 * 2 = 2 units squared.
Next, let's calculate the area of the rectangle. The base of the rectangle is 2 units (from x = 2 to x = 4) and the height is 1 unit. So the area of the rectangle is 2 * 1 = 2 units squared.
Therefore, the total area under the curve, which is the value of the integral from -1 to 4, is 2 + 2 = 4 units squared.
In the provided options, the closest answer to 4 is 4, so it seems that the answer is 4. However, if none of the options match exactly, it might be worth rechecking the calculations or seeking clarification from your instructor.
No problem! Let me help clarify the confusion.
To find the value of the integral of the piecewise linear function, you correctly separated the graph into triangles and rectangles. However, it seems there might be a misunderstanding in the calculations.
Let's analyze your two approaches:
1. Adding up the total area of each shape: This method is correct, but we need to double-check the calculations. Let's break it down step by step:
- Triangle 1: Base = 1, Height = 2, Area = (1 * 2) / 2 = 1
- Rectangle: Length = 2, Width = 2, Area = 2 * 2 = 4
- Triangle 2: Base = 1, Height = 1, Area = (1 * 1) / 2 = 0.5
Total area = Triangle 1 + Rectangle + Triangle 2 = 1 + 4 + 0.5 = 5.5
So, your first approach is correct. The total area is 5.5, not 4.
2. Subtracting the area below the x-axis: This approach is another valid method, but the calculation might have been incorrect. Let's recalculate it:
- Area below the x-axis: Triangle with base 2 and height 1, Area = (2 * 1) / 2 = 1
Total area above the x-axis = 5.5
Total area below the x-axis = 1 (as we just calculated)
Now, to find the net area, we subtract the area below the x-axis from the area above the x-axis:
Net area = Total area above the x-axis - Total area below the x-axis = 5.5 - 1 = 4.5
Therefore, the correct answer should be 4.5, not 4.
I hope this clears up the confusion and helps you determine the correct answer. If you have any further questions, feel free to ask!