Estimate the area under the graph of f(x)=cos((pi/4)x) from x=-2 to x=2 using 3 rectangles of equal width, using left and right endpoints.?
⊿ &rrtri;
To estimate the area under the graph of f(x) = cos((π/4)x) from x = -2 to x = 2 using 3 rectangles of equal width, we can use the left and right endpoints.
Step 1: Calculate the width of each rectangle.
The width of each rectangle can be calculated by dividing the total width by the number of rectangles. In this case, the total width is 2 - (-2) = 4, and we have 3 rectangles. So, the width of each rectangle is 4/3 = 1.3333.
Step 2: Calculate the heights of each rectangle.
For the left-endpoint estimation, we evaluate the function at the left endpoint of each rectangle and use that as the height. For the right-endpoint estimation, we evaluate the function at the right endpoint of each rectangle and use that as the height.
For the left-endpoint estimation:
The left endpoints of the 3 rectangles are -2, -1, and 0.
The height of the first rectangle is f(-2) = cos((π/4)(-2)) = cos(-π/2) = 0.
The height of the second rectangle is f(-1) = cos((π/4)(-1)) = cos(-π/4) = 0.7071.
The height of the third rectangle is f(0) = cos((π/4)(0)) = cos(0) = 1.
For the right-endpoint estimation:
The right endpoints of the 3 rectangles are -1, 0, and 1.
The height of the first rectangle is f(-1) = cos((π/4)(-1)) = cos(-π/4) = 0.7071.
The height of the second rectangle is f(0) = cos((π/4)(0)) = cos(0) = 1.
The height of the third rectangle is f(1) = cos((π/4)(1)) = cos(π/4) = 0.7071.
Step 3: Calculate the areas of each rectangle.
To calculate the area of each rectangle, multiply the width by the height.
For the left-endpoint estimation:
The area of the first rectangle is 1.3333 * 0 = 0.
The area of the second rectangle is 1.3333 * 0.7071 ≈ 0.9428.
The area of the third rectangle is 1.3333 * 1 = 1.3333.
For the right-endpoint estimation:
The area of the first rectangle is 1.3333 * 0.7071 ≈ 0.9428.
The area of the second rectangle is 1.3333 * 1 = 1.3333.
The area of the third rectangle is 1.3333 * 0.7071 ≈ 0.9428.
Step 4: Calculate the estimated area.
To calculate the estimated area, add up the areas of all the rectangles.
For the left-endpoint estimation:
Estimated area = 0 + 0.9428 + 1.3333 ≈ 2.275.
For the right-endpoint estimation:
Estimated area = 0.9428 + 1.3333 + 0.9428 ≈ 3.219.
Therefore, the estimated area under the graph of f(x) = cos((π/4)x) from x = -2 to x = 2 using 3 rectangles of equal width is approximately 2.275 using the left-endpoint estimation, and approximately 3.219 using the right-endpoint estimation.
To estimate the area under the graph of the function f(x) = cos((pi/4)x) from x = -2 to x = 2 using rectangles, we can use the left and right endpoints methods.
First, let's divide the interval from -2 to 2 into three equal subintervals. Since we have three rectangles, the width of each rectangle will be (2 - (-2))/3 = 4/3.
Now, let's calculate the left and right endpoints for each subinterval and evaluate the function at those points.
For the left endpoints:
- The left endpoint of the first subinterval is x = -2.
- The left endpoint of the second subinterval is x = -2 + (4/3) = -2/3.
- The left endpoint of the third subinterval is x = -2 + (8/3) = 2/3.
For the right endpoints:
- The right endpoint of the first subinterval is x = -2 + (4/3) = -2/3.
- The right endpoint of the second subinterval is x = -2 + (8/3) = 2/3.
- The right endpoint of the third subinterval is x = -2 + (12/3) = 2.
Let's calculate the function values at these endpoints:
- f(-2) = cos((pi/4)(-2)) = cos(-0.5π) ≈ 0.707
- f(-2/3) = cos((pi/4)(-2/3)) ≈ 0.383
- f(2/3) = cos((pi/4)(2/3)) ≈ 0.383
- f(2) = cos((pi/4)(2)) = cos(0.5π) = 0
Now, let's calculate the areas of the rectangles using the left and right endpoints:
For the left endpoints:
- Area of the first rectangle = (width) * (height) = (4/3) * 0.707 ≈ 0.942
- Area of the second rectangle = (4/3) * 0.383 ≈ 0.511
- Area of the third rectangle = (4/3) * 0.383 ≈ 0.511
For the right endpoints:
- Area of the first rectangle = (4/3) * 0.383 ≈ 0.511
- Area of the second rectangle = (4/3) * 0.383 ≈ 0.511
- Area of the third rectangle = (4/3) * 0 ≈ 0
Finally, to estimate the total area under the graph of f(x) from x = -2 to x = 2, we sum up the areas of the rectangles calculated using the left or right endpoints method:
For the left endpoints: 0.942 + 0.511 + 0.511 ≈ 1.964
For the right endpoints: 0.511 + 0.511 + 0 ≈ 1.022
Therefore, the estimated area under the graph of f(x) = cos((pi/4)x) from x = -2 to x = 2 using 3 rectangles of equal width, using left and right endpoints, is approximately 1.964 and 1.022, respectively.
The interval is of length 4, so you will have three rectangles of width 4/3.
So, what are the left (right) endpoints?
Calculate f(x) at those three values, and multiply by the rectangle width (4/3).