Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.
g(x) = 2x^2 − x − 1, [3, 5], 4 rectangles
__ < Area < __
the x-values are 3,3.5,4,4.5,5
Find y at those values and add the areas, starting from left or right.
What do you get?
To find the approximation of the area using left and right endpoints, we need to divide the interval [3,5] into four equal subintervals and use the left endpoints of each subinterval for one approximation and the right endpoints for the other approximation.
Step 1: Find the width of each subinterval.
The width of each subinterval is given by:
Width = (b - a) / n
where b is the upper limit of the interval, a is the lower limit of the interval, and n is the number of rectangles.
In our case:
b = 5
a = 3
n = 4
Width = (5 - 3) / 4 = 2 / 4 = 0.5
Step 2: Calculate the left endpoint values.
To find the left endpoint values, we start from the lower limit of the interval (a = 3) and increment by the width (0.5) for each rectangle.
First left endpoint value = 3
Second left endpoint value = 3 + 0.5 = 3.5
Third left endpoint value = 3.5 + 0.5 = 4
Fourth left endpoint value = 4 + 0.5 = 4.5
Step 3: Calculate the right endpoint values.
To find the right endpoint values, we start from the lower limit of the interval (a = 3) and increment by the width (0.5) for each rectangle.
First right endpoint value = 3 + 0.5 = 3.5
Second right endpoint value = 3.5 + 0.5 = 4
Third right endpoint value = 4 + 0.5 = 4.5
Fourth right endpoint value = 4.5 + 0.5 = 5
Step 4: Calculate the heights for each approximation.
To calculate the height for each approximation, substitute the left and right endpoint values into the function g(x) = 2x^2 − x − 1.
For the left endpoint approximation:
First height = g(3) = 2(3)^2 - 3 - 1 = 18 - 3 - 1 = 14
Second height = g(3.5) = 2(3.5)^2 - 3.5 - 1 = 24.5 - 3.5 - 1 = 20
Third height = g(4) = 2(4)^2 - 4 - 1 = 32 - 4 - 1 = 27
Fourth height = g(4.5) = 2(4.5)^2 - 4.5 - 1 = 40.5 - 4.5 - 1 = 35
For the right endpoint approximation:
First height = g(3.5) = 2(3.5)^2 - 3.5 - 1 = 24.5 - 3.5 - 1 = 20
Second height = g(4) = 2(4)^2 - 4 - 1 = 32 - 4 - 1 = 27
Third height = g(4.5) = 2(4.5)^2 - 4.5 - 1 = 40.5 - 4.5 - 1 = 35
Fourth height = g(5) = 2(5)^2 - 5 - 1 = 50 - 5 - 1 = 44
Step 5: Calculate the area for each approximation.
To calculate the area for each approximation, multiply the width of each rectangle by its corresponding height and sum up the results.
For the left endpoint approximation:
Area = Width * (First height + Second height + Third height + Fourth height)
= 0.5 * (14 + 20 + 27 + 35)
= 0.5 * 96
= 48
For the right endpoint approximation:
Area = Width * (First height + Second height + Third height + Fourth height)
= 0.5 * (20 + 27 + 35 + 44)
= 0.5 * 126
= 63
Therefore, the approximation for the area using left endpoints and using four rectangles is:
48 < Area < 63.
To find the area of the region between the graph of the function g(x) = 2x^2 − x − 1 and the x-axis over the interval [3, 5] using left and right endpoints, we need to divide the interval into rectangles and calculate the area of each rectangle.
First, let's divide the interval [3, 5] into four equal subintervals. Since we have four rectangles, each subinterval will have a width of (5 - 3) / 4 = 0.5.
Using the left endpoints method, we will choose the leftmost point of each subinterval as the x-coordinate for the height of the rectangle. Let's label the left endpoints as follows:
- For the first subinterval: x = 3
- For the second subinterval: x = 3.5
- For the third subinterval: x = 4
- For the fourth subinterval: x = 4.5
Now let's calculate the heights of the rectangles by plugging these x-values into the function g(x) = 2x^2 − x − 1:
- For the first rectangle: g(3) = 2(3)^2 - 3 - 1 = 16
- For the second rectangle: g(3.5) = 2(3.5)^2 - 3.5 - 1 = 23.25
- For the third rectangle: g(4) = 2(4)^2 - 4 - 1 = 31
- For the fourth rectangle: g(4.5) = 2(4.5)^2 - 4.5 - 1 = 40.75
Now, we can calculate the area of each rectangle by multiplying the height by the width (0.5):
- For the first rectangle: area = 16 * 0.5 = 8
- For the second rectangle: area = 23.25 * 0.5 = 11.625
- For the third rectangle: area = 31 * 0.5 = 15.5
- For the fourth rectangle: area = 40.75 * 0.5 = 20.375
To find the lower and upper approximations of the area, we can sum the areas of the rectangles:
- Lower approximation: sum of the areas of the rectangles using the left endpoints method = 8 + 11.625 + 15.5 + 20.375 = 55.5
- Upper approximation: sum of the areas of the rectangles using the right endpoints method = 11.625 + 15.5 + 20.375 + 25 = 72.5
Therefore, the lower approximation of the area is 55.5 and the upper approximation of the area is 72.5.
In summary, the area between the graph of the function g(x) = 2x^2 − x − 1 and the x-axis over the interval [3, 5] using four rectangles is 55.5 < Area < 72.5.