Estimate the area under the graph of f(x)=x^2+2x from x = 4 to x = 10 using 3 approximating rectangles and left endpoints.
I got 290.25 as my answer but it's wrong. Would someone explain to me how to answer this question.
Well, estimating the area under a curve using rectangles is quite the challenge. It can be a bit like playing Tetris, but with math. Let's see if I can help you out, and maybe inject a little humor along the way!
To estimate the area under the graph of f(x) = x^2 + 2x, we'll divide the interval from x = 4 to x = 10 into three equal subintervals. Since we're using left endpoints, we start by evaluating the function at the leftmost point of each subinterval.
For the first rectangle, the leftmost point is x = 4. Plugging that into our function, we get f(4) = 4^2 + 2(4) = 16 + 8 = 24. So the left height of the first rectangle is 24.
Moving on to the second rectangle, the leftmost point is x = 6. Evaluating f(6), we get f(6) = 6^2 + 2(6) = 36 + 12 = 48. So the left height of the second rectangle is 48.
Finally, for the third rectangle, the leftmost point is x = 8. Plugging that into our function, we get f(8) = 8^2 + 2(8) = 64 + 16 = 80. So the left height of the third rectangle is 80.
To estimate the area, we'll calculate the width of each rectangle by dividing the interval from 4 to 10 into three equal subintervals. Since (10 - 4)/3 = 6/3 = 2, each rectangle has a width of 2.
Multiplying the width and height of each rectangle, we find the areas:
First rectangle: 2 * 24 = 48
Second rectangle: 2 * 48 = 96
Third rectangle: 2 * 80 = 160
Now, add up all the areas to get the estimate:
48 + 96 + 160 = 304
Ta-da! Our estimate for the area under the graph is 304. Hopefully, that puts a smile on your face, even if the numbers didn't quite match up. Remember, math can be a bit tricky sometimes, but the important thing is to keep trying and have a good laugh along the way!
To estimate the area under the graph of the function f(x) = x^2 + 2x from x = 4 to x = 10 using 3 approximating rectangles and left endpoints, you can use the Left Endpoint Riemann Sum method.
1. Begin by dividing the interval [4, 10] into 3 equal sub-intervals. The width of each sub-interval will be (10 - 4) / 3 = 2.
The 3 sub-intervals will be:
[4, 6]
[6, 8]
[8, 10]
2. Now, calculate the height of each rectangle by evaluating the function at the left endpoint of each sub-interval.
For the first sub-interval [4, 6], the left endpoint is x = 4.
Height of the first rectangle = f(4) = (4^2) + (2 * 4) = 16 + 8 = 24.
For the second sub-interval [6, 8], the left endpoint is x = 6.
Height of the second rectangle = f(6) = (6^2) + (2 * 6) = 36 + 12 = 48.
For the third sub-interval [8, 10], the left endpoint is x = 8.
Height of the third rectangle = f(8) = (8^2) + (2 * 8) = 64 + 16 = 80.
3. Calculate the total area by multiplying the width of each rectangle by its corresponding height, and then summing up the areas.
Area of the first rectangle = (width) * (height) = 2 * 24 = 48.
Area of the second rectangle = 2 * 48 = 96.
Area of the third rectangle = 2 * 80 = 160.
Total area ≈ (48 + 96 + 160) = 304.
Therefore, the estimated area under the graph of f(x) = x^2 + 2x from x = 4 to x = 10 using 3 approximating rectangles and left endpoints is 304, not 290.25.
To estimate the area under a graph using rectangles, we divide the interval between the two given values of x into smaller subintervals and approximate the area under the graph over each subinterval using rectangles. In this case, we are given that we need to use 3 rectangles with left endpoints.
To find the width of each rectangle, we first calculate the width of each subinterval. The total width is the difference between the two given x-values:
Width of each subinterval = (10 - 4) / 3 = 6 / 3 = 2
Now, we need to choose the left endpoint of each subinterval to calculate the height of each rectangle. The left endpoints for the 3 rectangles are: 4, 6, and 8.
To calculate the height of each rectangle, we substitute these left endpoints into the function f(x) = x^2 + 2x:
Height of first rectangle = f(4) = (4)^2 + 2(4) = 16 + 8 = 24
Height of second rectangle = f(6) = (6)^2 + 2(6) = 36 + 12 = 48
Height of third rectangle = f(8) = (8)^2 + 2(8) = 64 + 16 = 80
Now, we calculate the area of each rectangle by multiplying the width by the height:
Area of first rectangle = Width * Height = 2 * 24 = 48
Area of second rectangle = Width * Height = 2 * 48 = 96
Area of third rectangle = Width * Height = 2 * 80 = 160
Finally, we sum the areas of all three rectangles to estimate the total area under the graph:
Estimated area = Sum of all rectangle areas = 48 + 96 + 160 = 304
So, the estimated area under the graph of f(x) = x^2 + 2x from x = 4 to x = 10 using three approximating rectangles and left endpoints is 304.
what is f(x) at x=4,6, 8
area=f(4)*2+f(6)*2 + f(8)*2
that is three rectangles of width 2, height to f(x)(left end point).
How in the world would you estimate to five digits of precision. "estimate"
f(4)=24
f(6)=48
f(8)=80
estimate: you do it. Twice the sum of those numbers.