Estimate the area under the graph of
f(x)=3x^3+5
from x=-1 to x=4, first using 5 approximating rectangles and right endpoints, and then improving your estimate using 10 approximating rectangles and right endpoints.
I'm having problem with using the midpoint to find the area. With 5 rectangle I have 210.625 but I can't figure out how to do the 10 rectangle.
again, you cannot estimate to six significant digits.
on the five rectangle, right endpoints, use 4,3,2,1,0 as right endpoints, width 1
area= f(4)*1 + f(3)*1 + ...
now for ten, use 4, 3.5, 3, 2.5....to -.5, then area width is .5
To estimate the area under the graph of a function using rectangles with right endpoints, you need to follow these steps:
1. Divide the interval [a, b] into equal subintervals based on the number of rectangles you want to use. In this case, you have x values ranging from -1 to 4, so the interval is [a, b] = [-1, 4].
For the first part of the question, you have 5 rectangular approximations, so you need to divide the interval into 5 subintervals. The width of each subinterval will be (4 - (-1))/5 = 5/5 = 1.
2. Identify the right endpoint of each subinterval. You start with the right endpoint of the first subinterval and move to the right until you reach the right endpoint of the last subinterval.
For the first part of the question, since you are using 5 rectangles, the right endpoints of the subintervals will be -1, 0, 1, 2, and 3.
3. Evaluate the function at each right endpoint to find the height of each rectangle. In other words, substitute the right endpoint values into the function and calculate the corresponding function values.
For the function f(x) = 3x^3 + 5, the function values at the right endpoints will be:
f(-1) = 3(-1)^3 + 5 = -2
f(0) = 3(0)^3 + 5 = 5
f(1) = 3(1)^3 + 5 = 8
f(2) = 3(2)^3 + 5 = 29
f(3) = 3(3)^3 + 5 = 98
4. Calculate the area of each rectangle by multiplying the width by the height. In this case, the width of each rectangle is 1 (as determined in step 1).
For the first part of the question, with 5 rectangles, the areas of the rectangles will be:
Rectangle 1: width * height = 1 * (-2) = -2
Rectangle 2: width * height = 1 * 5 = 5
Rectangle 3: width * height = 1 * 8 = 8
Rectangle 4: width * height = 1 * 29 = 29
Rectangle 5: width * height = 1 * 98 = 98
5. Sum up the areas of all the rectangles to get an estimate of the total area under the curve.
For the first part of the question, the estimate of the area under the graph using 5 rectangles and right endpoints is:
-2 + 5 + 8 + 29 + 98 = 138
To improve the estimate, you can repeat the process with a larger number of rectangles. For the second part of the question, you need to use 10 rectangles.
Repeat steps 1-5, but this time using 10 subintervals.
1. Divide the interval [a, b] = [-1, 4] into 10 equal subintervals.
The width of each subinterval will be (4 - (-1))/10 = 5/10 = 0.5.
2. Identify the right endpoints of each of the 10 subintervals.
The right endpoints will be: -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0.
3. Evaluate the function at each right endpoint to find the height of each rectangle.
This involves substituting the right endpoint values into the function and calculating the corresponding function values.
For f(x) = 3x^3 + 5, the function values at the right endpoints will be:
f(-1.0) = 3(-1.0)^3 + 5 = -2.0
f(-0.5) = 3(-0.5)^3 + 5 = 3.125
f(0.0) = 3(0.0)^3 + 5 = 5.0
f(0.5) = 3(0.5)^3 + 5 = 5.625
f(1.0) = 3(1.0)^3 + 5 = 8.0
f(1.5) = 3(1.5)^3 + 5 = 14.875
f(2.0) = 3(2.0)^3 + 5 = 29.0
f(2.5) = 3(2.5)^3 + 5 = 51.375
f(3.0) = 3(3.0)^3 + 5 = 98.0
f(3.5) = 3(3.5)^3 + 5 = 175.375
f(4.0) = 3(4.0)^3 + 5 = 293.0
4. Calculate the area of each rectangle by multiplying the width by the height.
In this case, the width of each rectangle is 0.5 (as determined in step 1).
For 10 rectangles, the areas of the rectangles will be:
Rectangle 1: width * height = 0.5 * (-2.0) = -1.0
Rectangle 2: width * height = 0.5 * 3.125 = 1.5625
Rectangle 3: width * height = 0.5 * 5.0 = 2.5
Rectangle 4: width * height = 0.5 * 5.625 = 2.8125
Rectangle 5: width * height = 0.5 * 8.0 = 4.0
Rectangle 6: width * height = 0.5 * 14.875 = 7.4375
Rectangle 7: width * height = 0.5 * 29.0 = 14.5
Rectangle 8: width * height = 0.5 * 51.375 = 25.6875
Rectangle 9: width * height = 0.5 * 98.0 = 49.0
Rectangle 10: width * height = 0.5 * 175.375 = 87.6875
5. Sum up the areas of all the rectangles to get an improved estimate of the total area under the curve.
For the second part of the question, the estimate of the area under the graph using 10 rectangles and right endpoints is:
-1.0 + 1.5625 + 2.5 + 2.8125 + 4.0 + 7.4375 + 14.5 + 25.6875 + 49.0 + 87.6875 = 205.1875.
Therefore, the improved estimate is 205.1875.