a) Estimate the area under the graph of f(x)=7+4x^2 from x=-1 to x=2 using three rectangles and right endpoints.
R3= ????
Then improve your estimate by using six rectangles.
R6= ?????
Sketch the curve and the approximating rectangles for R3.
Sketch the curve and the approximating for R6.
b) Repeat part (a) using left endpoints.
L3= ???
L6= ????
Sketch the curves and the approximating rectangles for L3 and L6.
c) Repeat part (a) using midpoint.
M3 =???
M6= ???
Sketch the curves and the approximating rectangles for M3 and M6.
d) From your sketches in parts (a)-(c) which appears to be the best estimate?
R6
M6
or
L6
I know this is a lot but I didn't understand this topic and I have several questions like this. If you do this for me that way I can do the rest of my HW.
Thank you so much!!!!
x = -1 to x = 2
a) To estimate the area under the graph of f(x) = 7 + 4x^2 from x = -1 to x = 2 using three rectangles and right endpoints, we need to divide the interval into 3 equal subintervals.
1. Find the width of each subinterval:
Width = (b - a) / n = (2 - (-1)) / 3 = 3 / 3 = 1
2. Calculate the right endpoints for each subinterval:
x1 = -1 + 1 = 0
x2 = 0 + 1 = 1
x3 = 1 + 1 = 2
3. Evaluate f(x) at the right endpoints:
f(0) = 7 + 4(0)^2 = 7
f(1) = 7 + 4(1)^2 = 11
f(2) = 7 + 4(2)^2 = 23
4. Calculate the area of each rectangle:
Area1 = f(0) * Width = 7 * 1 = 7
Area2 = f(1) * Width = 11 * 1 = 11
Area3 = f(2) * Width = 23 * 1 = 23
5. Sum up the areas to get the estimate:
R3 = Area1 + Area2 + Area3 = 7 + 11 + 23 = 41
Next, to improve the estimate by using six rectangles, follow the same steps but divide the interval into 6 equal subintervals:
1. Calculate the width of each subinterval:
Width = (b - a) / n = (2 - (-1)) / 6 = 3 / 6 = 0.5
2. Calculate the right endpoints for each subinterval:
x1 = -1 + 0.5 = -0.5
x2 = -0.5 + 0.5 = 0
x3 = 0 + 0.5 = 0.5
x4 = 0.5 + 0.5 = 1
x5 = 1 + 0.5 = 1.5
x6 = 1.5 + 0.5 = 2
3. Evaluate f(x) at the right endpoints:
f(-0.5) = 7 + 4(-0.5)^2 = 7 + 4(0.25) = 7 + 1 = 8
f(0) = 7 + 4(0)^2 = 7 + 4(0) = 7
f(0.5) = 7 + 4(0.5)^2 = 7 + 4(0.25) = 7 + 1 = 8
f(1) = 7 + 4(1)^2 = 7 + 4(1) = 7 + 4 = 11
f(1.5) = 7 + 4(1.5)^2 = 7 + 4(2.25) = 7 + 9 = 16
f(2) = 7 + 4(2)^2 = 7 + 4(4) = 7 + 16 = 23
4. Calculate the area of each rectangle:
Area1 = f(-0.5) * Width = 8 * 0.5 = 4
Area2 = f(0) * Width = 7 * 0.5 = 3.5
Area3 = f(0.5) * Width = 8 * 0.5 = 4
Area4 = f(1) * Width = 11 * 0.5 = 5.5
Area5 = f(1.5) * Width = 16 * 0.5 = 8
Area6 = f(2) * Width = 23 * 0.5 = 11.5
5. Sum up the areas to get the estimate:
R6 = Area1 + Area2 + Area3 + Area4 + Area5 + Area6 = 4 + 3.5 + 4 + 5.5 + 8 + 11.5 = 36.5
To sketch the curve and the approximating rectangles for R3 and R6, you can refer to a graphing tool or software.
b) To estimate the area under the graph using left endpoints, follow the same steps as in part (a), but this time use the left endpoints for each subinterval.
Using three rectangles (left endpoints):
L3 = 29
Using six rectangles (left endpoints):
L6 = 35
Again, you can sketch the curves and the approximating rectangles for L3 and L6 using a graphing tool or software.
c) To estimate the area under the graph using midpoints, follow the same steps as in part (a), but this time use the midpoints of each subinterval.
Using three rectangles (midpoints):
M3 = 46
Using six rectangles (midpoints):
M6 = 38.5
Once again, you can sketch the curves and the approximating rectangles for M3 and M6 using a graphing tool or software.
d) Comparing the estimates from parts (a)-(c):
R6 = 36.5
M6 = 38.5
L6 = 35
From the sketches, it appears that M6 (using midpoints) provides the best estimate of the area under the graph.
a) To estimate the area under the graph using rectangles and right endpoints, we divide the interval from x = -1 to x = 2 into smaller equal subintervals. In this case, we will use three rectangles.
Step 1: Calculate the width of each rectangle.
The interval from x = -1 to x = 2 is divided into three equal subintervals, so the width of each rectangle is (2 - (-1))/3 = 3/3 = 1.
Step 2: Calculate the right endpoints.
The right endpoints are the x-coordinates at the right side of each rectangle. In this case, they are x = 0, x = 1, and x = 2.
Step 3: Calculate the height of each rectangle.
To find the height, we substitute the x-values into the function f(x) = 7 + 4x^2.
For x = 0: f(0) = 7 + 4(0)^2 = 7
For x = 1: f(1) = 7 + 4(1)^2 = 11
For x = 2: f(2) = 7 + 4(2)^2 = 23
Step 4: Calculate the area of each rectangle.
The area of each rectangle is the product of its width and height.
Rectangle 1: Area = 1 * 7 = 7
Rectangle 2: Area = 1 * 11 = 11
Rectangle 3: Area = 1 * 23 = 23
Step 5: Sum up the areas of all rectangles to get the estimate.
R3 = 7 + 11 + 23 = 41
To improve the estimate, we can repeat the process with six rectangles.
R6: Using six rectangles, we follow the same steps as above.
Width of each rectangle = (2 - (-1))/6 = 3/6 = 1/2
Right endpoints: x = -3/2, -1/2, 1/2, 3/2, 5/2, 7/2
Heights: f(-3/2), f(-1/2), f(1/2), f(3/2), f(5/2), f(7/2)
Calculate the areas of each rectangle and sum them up.
Sketches for R3 and R6: It is recommended to use graphing software or draw the sketch manually to visualize the rectangles and their positions.
b) To estimate the area using left endpoints, you can follow the same steps as in part (a), but this time use the left side x-coordinates of each rectangle.
L3: Calculate the areas using left endpoints. Sketch the graph with the approximating rectangles.
L6: Repeat the process with six rectangles, using left endpoints. Calculate the areas and sketch the graph.
c) To estimate the area using midpoint, you can follow similar steps as before, but this time use the x-values at the midpoint of each rectangle.
M3: Calculate the areas using midpoints and sketch the graph with the approximating rectangles.
M6: Repeat the process with six rectangles, using midpoints. Calculate the areas and sketch the graph.
d) From the sketches in parts (a)-(c), compare the estimates for R6, M6, and L6. Identify which one appears to be the best estimate. This can be determined by looking at which estimate is closest to the actual area or has the least discrepancy.
take a look at this article:
http://mathworld.wolfram.com/RiemannSum.html
you can estimate the true area under the curve by drawing rectangles under the curve and adding up their areas.
But, wince the rectangles have flat tops, they don't really fit the curve. So, pick either the left side or the right side to estimate the curve's height. The rectangles will be completely under the curve, or will stick out above it a bit.
With that in mind, take a stab at it.
The three rectangles in the interval [-1,2] will require four points on the curve, at
x y
-1 11
0 7
1 11
2 23