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. (Round your answers to four decimal places.)
g(x) = 4 sin x, [0, 𝜋], 6 rectangles
left endpoints approximation Area:
right endpoints approximation Area:
Why did the function go to the circus? Because it wanted to find some rectangular fun!
Using left endpoints, let's approximate the area of the region between the graph of g(x) = 4 sin x and the x-axis over the interval [0, 𝜋], with 6 rectangles.
First, let's divide the interval [0, 𝜋] into 6 equal subintervals. Each subinterval will have a width of (𝜋 - 0)/6 = 𝜋/6.
Using the left endpoints of each subinterval, we can calculate the height of the rectangle by evaluating g(x) at the left endpoint.
For the first rectangle: x = 0
Height: g(0) = 4 sin 0 = 0
For the second rectangle: x = 𝜋/6
Height: g(𝜋/6) = 4 sin(𝜋/6) ≈ 1.7321
For the third rectangle: x = 2𝜋/6
Height: g(2𝜋/6) = 4 sin(2𝜋/6) ≈ 3.4641
And so on, until the sixth rectangle: x = 5𝜋/6
Height: g(5𝜋/6) = 4 sin(5𝜋/6) ≈ 3.4641
Now, we can calculate the area of each rectangle by multiplying the width and height.
For the first rectangle: Area = (𝜋/6) * 0 = 0
For the second rectangle: Area = (𝜋/6) * 1.7321 ≈ 0.9072
For the third rectangle: Area = (𝜋/6) * 3.4641 ≈ 1.8138
And so on, until the sixth rectangle: Area = (𝜋/6) * 3.4641 ≈ 1.8138
To find the left endpoints approximation, we add up the areas of all the rectangles:
Approximation Area ≈ 0 + 0.9072 + 1.8138 + 1.8138 + 1.8138 + 1.8138 ≈ 7.2624
Now, let's move to the right endpoints approximation. The process is similar, but this time we will use the right endpoints of each subinterval to calculate the height of the rectangle.
For the first rectangle: x = 𝜋/6
Height: g(𝜋/6) ≈ 1.7321
For the second rectangle: x = 2𝜋/6
Height: g(2𝜋/6) ≈ 3.4641
For the third rectangle: x = 3𝜋/6
Height: g(3𝜋/6) ≈ 4.8989
And so on, until the sixth rectangle: x = 𝜋
Height: g(𝜋) = 4 sin 𝜋 = 0
Now, we can calculate the area of each rectangle by multiplying the width and height.
For the first rectangle: Area = (𝜋/6) * 1.7321 ≈ 0.9072
For the second rectangle: Area = (𝜋/6) * 3.4641 ≈ 1.8138
For the third rectangle: Area = (𝜋/6) * 4.8989 ≈ 2.5669
And so on, until the sixth rectangle: Area = (𝜋/6) * 0 = 0
To find the right endpoints approximation, we add up the areas of all the rectangles:
Approximation Area ≈ 0.9072 + 1.8138 + 2.5669 + 2.5669 + 2.5669 + 0 ≈ 10.4217
To find the left and right endpoints approximations of the area between the graph of the function g(x) = 4 sin x and the x-axis over the interval [0, π] using 6 rectangles, follow these steps:
Step 1: Calculate the width of each rectangle.
The interval [0, π] will be divided into 6 equal subintervals, so the width of each rectangle will be the length of each subinterval. To calculate, we need to find the difference between the upper and lower limits of the interval and divide it by the number of rectangles.
Width of each rectangle = (π - 0) / 6 = π/6
Step 2: Calculate the height of each rectangle.
Take the left endpoint of each subinterval and evaluate the function g(x) = 4 sin x at that point. This will give us the height of each rectangle.
Left endpoints approximation:
For the left endpoints approximation, we evaluate the function at the left endpoint of each subinterval:
Height of each rectangle = g(0), g(π/6), g(2π/6), g(3π/6), g(4π/6), g(5π/6)
= 4sin(0), 4sin(π/6), 4sin(2π/6), 4sin(3π/6), 4sin(4π/6), 4sin(5π/6)
Step 3: Calculate the area of each rectangle.
The area of each rectangle is found by multiplying the width by the height.
Area of each rectangle = Width * Height
Step 4: Sum up the areas of all the rectangles to find the total approximation.
Left endpoints approximation:
Add up the areas of all 6 rectangles to find the total approximation of the area.
Area = Sum of the areas of each rectangle = Area1 + Area2 + Area3 + ... + Area6
Repeat Steps 2-4 for the right endpoints approximation.
Right endpoints approximation:
For the right endpoints approximation, we evaluate the function at the right endpoint of each subinterval:
Height of each rectangle = g(π/6), g(2π/6), g(3π/6), g(4π/6), g(5π/6), g(π)
Repeat Steps 3-4 to find the right endpoints approximation.
Now let's calculate the left and right endpoints approximations.
Step 1: Calculate the width of each rectangle.
Width of each rectangle = π/6
Step 2: Calculate the height of each rectangle.
Left endpoints approximation:
Height of each rectangle = 4sin(0), 4sin(π/6), 4sin(2π/6), 4sin(3π/6), 4sin(4π/6), 4sin(5π/6)
= 0, 4sin(π/6), 4sin(π/3), 4sin(π/2), 4sin(2π/3), 4sin(5π/6)
Right endpoints approximation:
Height of each rectangle = 4sin(π/6), 4sin(2π/6), 4sin(3π/6), 4sin(4π/6), 4sin(5π/6), 4sin(π)
= 4sin(π/6), 4sin(π/3), 4sin(π/2), 4sin(2π/3), 4sin(5π/6), 0
Step 3: Calculate the area of each rectangle.
Area of each rectangle = Width * Height
Left endpoints approximation:
Area1 = (π/6) * 0
Area2 = (π/6) * 4sin(π/6)
Area3 = (π/6) * 4sin(π/3)
Area4 = (π/6) * 4sin(π/2)
Area5 = (π/6) * 4sin(2π/3)
Area6 = (π/6) * 4sin(5π/6)
Right endpoints approximation:
Area1 = (π/6) * 4sin(π/6)
Area2 = (π/6) * 4sin(π/3)
Area3 = (π/6) * 4sin(π/2)
Area4 = (π/6) * 4sin(2π/3)
Area5 = (π/6) * 4sin(5π/6)
Area6 = (π/6) * 0
Step 4: Sum up the areas of all the rectangles to find the total approximation.
Left endpoints approximation:
Area = Area1 + Area2 + Area3 + Area4 + Area5 + Area6
Right endpoints approximation:
Area = Area1 + Area2 + Area3 + Area4 + Area5 + Area6
Finally, round the answers to four decimal places.
Left endpoints approximation Area: Sum of the areas of each rectangle (rounded to four decimal places)
Right endpoints approximation Area: Sum of the areas of each rectangle (rounded to four decimal places)
To find the left and right endpoints approximations of the area of the region between the graph of the function and the x-axis, we can divide the interval [0, 𝜋] into smaller subintervals based on the number of rectangles given, which is 6 in this case.
Step 1: Determine the width of each rectangle.
The width of each rectangle will be the width of the interval divided by the number of rectangles:
width of each rectangle = (𝜋 - 0) / 6 = 𝜋 / 6
Step 2: Calculate the left and right endpoints of each subinterval.
Left endpoint of each subinterval: The left endpoint of the first subinterval is the left endpoint of the interval, which is 0. For the subsequent subintervals, the left endpoint can be found by adding the width of each rectangle to the previous left endpoint.
right endpoint of each subinterval: The right endpoint of the first subinterval is the left endpoint of the first subinterval plus the width of each rectangle. For the subsequent subintervals, the right endpoint can be found by adding the width of each rectangle to the previous right endpoint.
Step 3: Evaluate the function at each left and right endpoint to find the height of each rectangle.
For each subinterval, evaluate the function at the left endpoint to find the height of the rectangle for the left endpoints approximation. Similarly, evaluate the function at the right endpoint to find the height of the rectangle for the right endpoints approximation.
Step 4: Calculate the area of each rectangle.
The area of each rectangle is given by the product of its width and height.
Step 5: Sum up the areas of all the rectangles to find the left and right endpoints approximations of the area.
Let's go through the steps to find the left and right endpoints approximations of the area using the given information:
Step 1: width of each rectangle = 𝜋 / 6
Step 2:
For the left endpoints:
Subinterval 1: [0, 𝜋 / 6]
Subinterval 2: [𝜋 / 6, 2𝜋 / 6]
Subinterval 3: [2𝜋 / 6, 3𝜋 / 6]
Subinterval 4: [3𝜋 / 6, 4𝜋 / 6]
Subinterval 5: [4𝜋 / 6, 5𝜋 / 6]
Subinterval 6: [5𝜋 / 6, 𝜋]
For the right endpoints:
Subinterval 1: [𝜋 / 6, 2𝜋 / 6]
Subinterval 2: [2𝜋 / 6, 3𝜋 / 6]
Subinterval 3: [3𝜋 / 6, 4𝜋 / 6]
Subinterval 4: [4𝜋 / 6, 5𝜋 / 6]
Subinterval 5: [5𝜋 / 6, 6𝜋 / 6]
Subinterval 6: [6𝜋 / 6, 7𝜋 / 6]
Step 3:
For each subinterval, evaluate the function g(x) = 4 sin x at the left and right endpoints to find the height of each rectangle.
Step 4:
Calculate the area of each rectangle by multiplying its width and height.
Step 5:
Sum up the areas of all the rectangles to find the left and right endpoints approximations of the area.
Once you follow these steps, you will find the left endpoints approximation area and the right endpoints approximation area for the given function and interval.
the left endpoints are at x = 0, 𝜋/6, ... 5𝜋/6
so the area is
4*𝜋/6 (f(0) + f(𝜋/6) + ... + f(5𝜋/6)) = 4*𝜋/6 (0 + 1/2 + √3/2 + 1 + √3/2 + 1/2) = 4*𝜋/6 (2+√3)
the right endpoints are the same, since f(𝜋) = f(0)