what is the exact area of f(x)= 4x+5 from x=1 to x=4 using the limit of an infinite sum method.
To find the exact area of the function f(x) = 4x + 5 from x = 1 to x = 4 using the limit of an infinite sum method, we can express it as a Riemann sum.
The Riemann sum divides the interval [1, 4] into smaller partitions and approximates the area under the curve by summing the areas of rectangular strips.
First, let's define the width of each partition. We'll divide the interval into n equal subintervals, where n represents the number of partitions. In this case, each subinterval will have a width of Δx = (4 - 1) / n.
Next, we need to find the height of each rectangular strip. We can do this by evaluating the function f(x). For a given partition, the height of the strip will be f(xi), where xi represents the midpoint of each subinterval.
The Riemann sum is given by the sum of the areas of all these rectangular strips defined as:
Riemann sum = Σ[f(xi) * Δx]
To find the exact area, we need to take the limit as n approaches infinity:
Area = lim(n→∞) Σ[f(xi) * Δx]
Now, let's compute the Riemann sum step-by-step:
1. Determine the width of each subinterval: Δx = (4 - 1) / n.
2. Compute the midpoint of each subinterval: xi = 1 + (i - 0.5) * Δx, where i ranges from 1 to n.
3. Evaluate the function f(xi) at each midpoint.
4. Multiply each function value by the width of the subinterval to get the area of each rectangular strip.
5. Sum up all the areas of the rectangular strips.
6. Take the limit as n approaches infinity to get the exact area.
Note that as we increase the number of partitions, n, the Riemann sum becomes a better approximation of the exact area.
Using this method, you can find the exact area of any function within a given interval.