Approximate the area under the curve on the given interval using n rectangles and the evaluation rules (a)left endpoint, (b) midpoint and (c) right endpoint.
y=(x+2)^(1/2) on [1,4], n=16
How am I suppose to do this?
It looks like they want you to use the trapezoidal rule, which is explained here:
http://metric.ma.ic.ac.uk/integration/techniques/definite/numerical-methods/trapezoidal-rule/
Looks more primitive than trapezoids to me, rectangles.
In each interval, use the y value on the left of that block for the height in part a
In each interval use the y value on the right of that block for part c (you have already calculated most of them for part a because except for the first one on the left, the left in part a becomes the right in part c
Now part b is a pain because you have to calculate new values of y in the middle of each interval and for the area of that block use that height times the interval in x
To approximate the area under the curve using n rectangles and the evaluation rules, you can follow these steps:
Step 1: Determine the width of each rectangle.
- Calculate the width of each rectangle by dividing the interval width (b - a) by the number of rectangles (n).
- In this case, the interval is [1, 4] and the number of rectangles is 16. So the width of each rectangle (Δx) is (4 - 1) / 16 = 3/16.
Step 2: Choose an evaluation rule.
- In this case, you need to use three different evaluation rules: (a) left endpoint, (b) midpoint, and (c) right endpoint.
- Each rule determines the height of the rectangle in a specific way.
Step 3: Calculate the heights of the rectangles using the chosen evaluation rule.
(a) Left Endpoint Rule:
- For each rectangle, evaluate the function at the left endpoint of the subinterval and use that as the height.
- Start from the left endpoint (a = 1) and evaluate the function at each left endpoint of the subinterval (a + i * Δx), where i is the index of the rectangle.
- For this case, evaluate the function at 1 + i * (3/16) for i = 0 to 15.
(b) Midpoint Rule:
- For each rectangle, evaluate the function at the midpoint of the subinterval and use that as the height.
- Start from the midpoint of the first subinterval and evaluate the function at each midpoint (a + (i + 0.5) * Δx), where i is the index of the rectangle.
- For this case, evaluate the function at 1 + (i + 0.5) * (3/16) for i = 0 to 15.
(c) Right Endpoint Rule:
- For each rectangle, evaluate the function at the right endpoint of the subinterval and use that as the height.
- Start from the right endpoint (b = 4) and evaluate the function at each right endpoint of the subinterval (a + (i + 1) * Δx), where i is the index of the rectangle.
- For this case, evaluate the function at 1 + (i + 1) * (3/16) for i = 0 to 15.
Step 4: Calculate the area of each rectangle.
- Multiply the width of each rectangle (Δx) by its corresponding height calculated in Step 3.
Step 5: Sum up the areas of all rectangles to get the approximate area under the curve.
I hope this helps! Let me know if you have any further questions.
To approximate the area under the curve using rectangles and evaluation rules, you can follow these steps:
1. Determine the width of each rectangle: Divide the interval [1, 4] into n subintervals, where n is the number of rectangles you want to use. In this case, n = 16. So, the width of each rectangle will be (4 - 1) / 16 = 0.1875.
2. Determine the x-values for each rectangle: For the left endpoint rule, start with the left endpoint of the interval (in this case, 1), and increment by the width of each rectangle. For the midpoint rule, start with the left endpoint plus half the width of each rectangle, and increment by the width. For the right endpoint rule, start with the left endpoint plus the width of each rectangle, and increment by the width.
3. Evaluate the function at each x-value: For each x-value obtained in step 2, plug it into the function y = (x+2)^(1/2) to get the corresponding y-value.
4. Calculate the area of each rectangle: Multiply the width of each rectangle by its corresponding y-value obtained in step 3.
5. Sum up the areas of all the rectangles: Add up all the areas calculated in step 4 to get the approximate area under the curve.
For the left endpoint rule:
- Start with x = 1 as the left endpoint.
- Calculate the y-values by plugging each x-value into the function: y = (x+2)^(1/2).
- Multiply each width by its corresponding y-value.
- Sum up all the areas obtained to get the approximate area.
For the midpoint rule:
- Start with x = 1 + 0.1875/2 = 1.09375 as the first x-value.
- Calculate the y-values by plugging each x-value into the function: y = (x+2)^(1/2).
- Multiply each width by its corresponding y-value.
- Sum up all the areas obtained to get the approximate area.
For the right endpoint rule:
- Start with x = 1 + 0.1875 = 1.1875 as the first x-value.
- Calculate the y-values by plugging each x-value into the function: y = (x+2)^(1/2).
- Multiply each width by its corresponding y-value.
- Sum up all the areas obtained to get the approximate area.
By following these steps, you can approximate the area under the curve using rectangles and the evaluation rules.