a) Estimate the area under the graph of
f(x) = 10(sqrt(x))
from x = 0 to x = 4 using four approximating rectangles and right endpoints
just figure f(1)...f(4)
since the rectangles have width 1, your approximation is just the sum of the four values of f.
The value will be a little high, since f is concave downward, so using the right ends of the intervals means the rectangles are a bit too tall.
To estimate the area under the graph of the function f(x) = 10(sqrt(x)) from x = 0 to x = 4 using four approximating rectangles and right endpoints, follow these steps:
1. Divide the interval [0, 4] into four equal subintervals with a width of Δx = (4 - 0) / 4 = 1.
2. Determine the x-coordinates of the right endpoints for each subinterval. In this case, the right endpoints are 1, 2, 3, and 4.
3. Evaluate the function f(x) = 10(sqrt(x)) at each right endpoint to get the corresponding heights of the rectangles.
For x = 1, f(1) = 10(sqrt(1)) = 10(1) = 10.
For x = 2, f(2) = 10(sqrt(2)) ≈ 10(1.41) ≈ 14.1.
For x = 3, f(3) = 10(sqrt(3)) ≈ 10(1.73) ≈ 17.3.
For x = 4, f(4) = 10(sqrt(4)) = 10(2) = 20.
4. Multiply the width Δx by the height of each rectangle to find their respective areas.
For the first rectangle, the area is 1 * 10 = 10.
For the second rectangle, the area is 1 * 14.1 ≈ 14.1.
For the third rectangle, the area is 1 * 17.3 ≈ 17.3.
For the fourth rectangle, the area is 1 * 20 = 20.
5. Add up the areas of all four rectangles to estimate the total area under the graph.
Total area ≈ 10 + 14.1 + 17.3 + 20 = 61.4.
Therefore, the estimated area under the graph of f(x) = 10(sqrt(x)) from x = 0 to x = 4 using four approximating rectangles and right endpoints is approximately 61.4.
To estimate the area under the graph of a function using rectangles, we can use a technique called Riemann sums. In this case, we'll use right endpoints to determine the heights of the rectangles.
Step 1: Divide the interval [0, 4] into four equal subintervals.
Since we want to use four approximating rectangles, we'll divide the interval [0, 4] into four subintervals of equal width.
First subinterval: [0, 1]
Second subinterval: [1, 2]
Third subinterval: [2, 3]
Fourth subinterval: [3, 4]
Step 2: Determine the height of each rectangle.
To find the height of each rectangle, we'll evaluate the function at the right endpoint of each subinterval.
For the first subinterval [0, 1], the right endpoint is 1. So the height of the first rectangle is:
f(1) = 10√(1) = 10(1) = 10
For the second subinterval [1, 2], the right endpoint is 2. So the height of the second rectangle is:
f(2) = 10√(2) = 10√(2)
For the third subinterval [2, 3], the right endpoint is 3. So the height of the third rectangle is:
f(3) = 10√(3) = 10√(3)
For the fourth subinterval [3, 4], the right endpoint is 4. So the height of the fourth rectangle is:
f(4) = 10√(4) = 10(2) = 20
Step 3: Calculate the area of each rectangle.
The width of each rectangle is the same since we divided the interval into equally spaced subintervals. In this case, the width of each rectangle is 1.
For the first rectangle, the area is:
10 * 1 = 10
For the second rectangle, the area is:
10√(2) * 1 = 10√(2)
For the third rectangle, the area is:
10√(3) * 1 = 10√(3)
For the fourth rectangle, the area is:
20 * 1 = 20
Step 4: Calculate the total estimated area.
To find the estimated area under the curve, we add up the areas of all four rectangles.
Total estimated area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4
Total estimated area = 10 + 10√(2) + 10√(3) + 20
Simplifying the expression will give you the estimated area under the curve from x = 0 to x = 4 using four approximating rectangles and right endpoints.