Estimate the area under the graph of f(x) = 2x^3 + 3 from x= -1 to x=5, first using 6 approximating rectangle and right endpoints, and then improving your estimate using 12 approximating rectangles and right endpoints. When using the left and right endpoints, is there a different way to compute each?
To estimate the area under the graph of f(x) = 2x^3 + 3 using approximating rectangles and right endpoints, we first need to divide the interval [-1, 5] into equal subintervals.
Case 1: 6 Approximating Rectangles
If we use 6 approximating rectangles, the width of each rectangle would be (5 - (-1))/6 = 1.2. We will evaluate the function at the right endpoints of each subinterval.
The x-values for the right endpoints are:
x₀ = -1 + 1.2 = 0.2
x₁ = 0.2 + 1.2 = 1.4
x₂ = 1.4 + 1.2 = 2.6
x₃ = 2.6 + 1.2 = 3.8
x₄ = 3.8 + 1.2 = 5.0
Now, let's calculate the approximate area under the curve using right endpoints with 6 rectangles:
Area = f(x₀) * 1.2 + f(x₁) * 1.2 + f(x₂) * 1.2 + f(x₃) * 1.2 + f(x₄) * 1.2
= (2(0.2)^3 + 3) * 1.2 + (2(1.4)^3 + 3) * 1.2 + (2(2.6)^3 + 3) * 1.2 + (2(3.8)^3 + 3) * 1.2 + (2(5)^3 + 3) * 1.2
Evaluate the above expression to get the numerical value of the area.
Case 2: 12 Approximating Rectangles
If we use 12 approximating rectangles, the width of each rectangle would be (5 - (-1))/12 = 0.6. Again, we will evaluate the function at the right endpoints of each subinterval.
The x-values for the right endpoints are:
x₀ = -1 + 0.6 = -0.4
x₁ = -0.4 + 0.6 = 0.2
x₂ = 0.2 + 0.6 = 0.8
x₃ = 0.8 + 0.6 = 1.4
x₄ = 1.4 + 0.6 = 2.0
x₅ = 2.0 + 0.6 = 2.6
x₆ = 2.6 + 0.6 = 3.2
x₇ = 3.2 + 0.6 = 3.8
x₈ = 3.8 + 0.6 = 4.4
x₉ = 4.4 + 0.6 = 5
Now, let's calculate the approximate area under the curve using right endpoints with 12 rectangles:
Area = f(x₀) * 0.6 + f(x₁) * 0.6 + f(x₂) * 0.6 + f(x₃) * 0.6 + f(x₄) * 0.6 + f(x₅) * 0.6
+ f(x₆) * 0.6 + f(x₇) * 0.6 + f(x₈) * 0.6 + f(x₉) * 0.6 + f(x₁₀) * 0.6 + f(x₁₁) * 0.6
Evaluate the above expression to get the numerical value of the area.
When using the left and right endpoints, there is indeed a different way to compute each. For the right endpoints, we evaluate the function at the rightmost x-value of each subinterval. In contrast, for the left endpoints, we would evaluate the function at the leftmost x-value of each subinterval.
To estimate the area under the graph of a function, you can use a method called Riemann sum. This involves dividing the interval between two points into smaller subintervals and approximating the area within each subinterval using rectangles.
To estimate the area under the graph of f(x) = 2x^3 + 3 from x = -1 to x = 5 using right endpoints, you can follow these steps:
Step 1: Determine the width of each subinterval.
In this case, you are using 6 approximating rectangles. The width of each subinterval can be calculated by dividing the total width of the interval (5 - (-1)) by the number of rectangles, which is 6. So, the width of each subinterval is (5 - (-1))/6 = 1.
Step 2: Calculate the height of each rectangle.
To find the height of each rectangle, you need to evaluate the function at the right endpoint of each subinterval. Since you are using right endpoints, you take the x-value at the right end of each subinterval.
For the first approximating rectangle with right endpoint x = -1 + 1 = 0, the value of f(x) is f(0) = 2(0)^3 + 3 = 3.
For the second approximating rectangle with right endpoint x = 0 + 1 = 1, the value of f(x) is f(1) = 2(1)^3 + 3 = 5.
You repeat this process for all 6 rectangles.
Step 3: Calculate the area of each rectangle.
To calculate the area of each rectangle, multiply the width of the rectangle by its height.
For the first rectangle, the area is 1 * 3 = 3.
For the second rectangle, the area is 1 * 5 = 5.
Continue this computation for all 6 rectangles.
Step 4: Add up the areas of all the rectangles.
To estimate the area under the graph, sum up the areas of all the rectangles. In this case, add up the areas of the 6 rectangles you've calculated.
3 + 5 + ... (continue this addition for all 6 rectangles) = total estimated area.
To improve the estimate, you can repeat these steps using a higher number of approximating rectangles. For example, if you use 12 approximating rectangles, follow the same steps as above but with a smaller width for each subinterval (total width divided by 12 instead of 6).
When using left and right endpoints, there is a different way to compute the height of each rectangle. For left endpoints, you evaluate the function at the left end of each subinterval, whereas for the right endpoints, you evaluate the function at the right end of each subinterval. The rest of the steps remain the same.
You are being asked to perform a numerical integration. Add up the values of f(x) at x= 1,2,3,4,5 and 6 . Multiply the sum by the interval, 1.
Then do it again at x= 1/2, 1, 3/2, 2,. ...6, and multiply the sum by the interval, 0.5.
If you use the average of left and right endpoints, you will be using the trapezoidal rule. It should be more accurate approximation to the integreal