Estimate the area under the graph of f(x)= x^2 + 2 x from x=1 to x=5 using 4 approximating rectangles and left endpoints.
I assume you want the rectangles to have equal widths. Tat would be (5-1)/4 = 1.
So, the left sides of the rectangles are at x = 1,2,3,4
the heights are f(1),f(2),f(3),f(4).
So, evaluate f there and add up the areas.
I got 50 and it's correct. Thank you. I didn't know how to find the 4 approximate values
Why did the rectangle go to the comedy club?
To find some area under the curve and have a laugh, of course!
Now, let me calculate the approximate area under the graph using left endpoints and 4 rectangles.
Since we have 4 rectangles, we divide the interval [1, 5] into 4 equal subintervals. Each subinterval has a width of (5 - 1)/4 = 1.
Now, let's calculate the left endpoint values for each subinterval:
For the first rectangle, the left endpoint is 1.
For the second rectangle, the left endpoint is 2.
For the third rectangle, the left endpoint is 3.
For the fourth rectangle, the left endpoint is 4.
Next, we calculate the height of each rectangle using the function f(x) = x^2 + 2x:
For the first rectangle, the height is f(1) = 1^2 + 2(1) = 3.
For the second rectangle, the height is f(2) = 2^2 + 2(2) = 10.
For the third rectangle, the height is f(3) = 3^2 + 2(3) = 15.
For the fourth rectangle, the height is f(4) = 4^2 + 2(4) = 24.
Now, we multiply the width and height of each rectangle to find their respective areas:
The area of the first rectangle is (1)(3) = 3.
The area of the second rectangle is (1)(10) = 10.
The area of the third rectangle is (1)(15) = 15.
The area of the fourth rectangle is (1)(24) = 24.
Finally, we add up the areas of all the rectangles to find the approximate area under the curve:
The total area is 3 + 10 + 15 + 24 = 52.
So, the approximate area under the graph of f(x) = x^2 + 2x from x = 1 to x = 5 using 4 approximating rectangles and left endpoints is 52.
To estimate the area under the graph of the function f(x) = x^2 + 2x from x=1 to x=5 using 4 approximating rectangles and left endpoints, follow these steps:
Step 1: Determine the width of each rectangle.
Since there are 4 rectangles, the width of each rectangle is equal to the total interval divided by the number of rectangles.
In this case, the total interval is 5 - 1 = 4, and the number of rectangles is 4.
So, the width of each rectangle is 4/4 = 1.
Step 2: Calculate the left endpoints.
The left endpoints are the x-values at the left side of each rectangle.
Starting from x=1, the left endpoints for the 4 rectangles would be: 1, 2, 3, and 4.
Step 3: Evaluate the function at each left endpoint.
Substitute each left endpoint into the function f(x) = x^2 + 2x to get the corresponding y-values.
For the left endpoints 1, 2, 3, and 4:
f(1) = (1)^2 + 2(1) = 1 + 2 = 3
f(2) = (2)^2 + 2(2) = 4 + 4 = 8
f(3) = (3)^2 + 2(3) = 9 + 6 = 15
f(4) = (4)^2 + 2(4) = 16 + 8 = 24
Step 4: Calculate the area of each rectangle.
The area of each rectangle is equal to the width multiplied by the corresponding y-value.
For the 4 rectangles starting from x=1:
Rectangle 1: Area = 1 * 3 = 3
Rectangle 2: Area = 1 * 8 = 8
Rectangle 3: Area = 1 * 15 = 15
Rectangle 4: Area = 1 * 24 = 24
Step 5: Sum up the areas of all the rectangles.
Add up the areas of all 4 rectangles to get the estimate for the total area under the graph.
3 + 8 + 15 + 24 = 50
Therefore, the estimated area under the graph of f(x) = x^2 + 2x from x=1 to x=5, using 4 approximating rectangles and left endpoints, is equal to 50 square units.
To estimate the area under the graph of f(x) = x^2 + 2x from x = 1 to x = 5 using 4 approximating rectangles and left endpoints, we can use the Left Riemann Sum method.
The Left Riemann Sum divides the interval [1, 5] into 4 equal subintervals, each with a width of (5 - 1) / 4 = 1.
To calculate the height of each rectangle, we evaluate the function f(x) at the left endpoint of each interval. So, the left endpoints are x = 1, 2, 3, and 4.
Using these left endpoints, we can calculate the height of each rectangle as follows:
- For x = 1, f(1) = 1^2 + 2*1 = 3.
- For x = 2, f(2) = 2^2 + 2*2 = 10.
- For x = 3, f(3) = 3^2 + 2*3 = 21.
- For x = 4, f(4) = 4^2 + 2*4 = 32.
The area of each rectangle is equal to the height multiplied by the width. Since the width of each rectangle is 1, the area of each rectangle is the same as its height.
Calculating the area of each rectangle, we have:
- Rectangle 1: height = 3.
- Rectangle 2: height = 10.
- Rectangle 3: height = 21.
- Rectangle 4: height = 32.
To estimate the total area under the graph, we sum up the areas of these 4 rectangles:
Area ≈ (3 + 10 + 21 + 32) * 1 = 66.
Therefore, the estimated area under the graph of f(x) = x^2 + 2x from x = 1 to x = 5, using 4 approximating rectangles and left endpoints, is approximately 66.