Use this definition with the right endpoints to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.
f(x)= 3+sin^2(x) 0<=x<=pi
Definition provided: The area of A of region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.
Work done: a=0 b=pi
change of x= (pi-0)/n
(I don't know how to format the rest of my work properly on here, so I guess I'll just say that the answer I got within the summation was 3+sin^2(i*(pi/n)
I have no idea why I'm wrong, could I have some help?
consider the area as a set of rectangles of width π/n
The right endpoints are iπ/n for i=1..n
So the area is
n
∑ (3+sin^2(iπ/n))(π/n)
i=1
I guess you forgot to multiply by the width of the rectangles. You just added up the heights.
Well, it seems like you're on the right track but made a small mistake in your calculation. Let's go through it step by step to find the correct expression for the area under the graph.
First, let's find the width of each rectangle. In this case, the interval is from 0 to π, so the width of each rectangle is (π - 0)/n, which simplifies to π/n.
Now, let's find the height of each rectangle. We know that the height is given by the function f(x) = 3 + sin^2(x). To get the height, we evaluate this function at the left endpoint of each rectangle.
So, for the i-th rectangle, the left endpoint is x = i*(π/n), and the height is f(i*(π/n)). Therefore, the expression for the area of each rectangle is:
A_i = (π/n) * [3 + sin^2(i*(π/n))]
Finally, to find the total area under the graph, we need to take the limit as n goes to infinity of the sum of these rectangles. This can be expressed as:
A = lim(n→∞) Σ[ i=1 to n ] (π/n) * [3 + sin^2(i*(π/n))]
Remember, we haven't evaluated the limit yet; we are just expressing it as a limit of the sum of the areas of approximating rectangles.
I hope this helps! Let me know if you have any more questions.
To find the expression for the area under the graph of f using the provided definition, you need to use the Riemann sum.
Here's how you can do it step-by-step:
1. Determine the interval of integration: Given the function f(x) = 3 + sin^2(x) on the interval 0 <= x <= pi, the interval of integration is a = 0 and b = pi.
2. Partition the interval: Divide the interval [a, b] into n equal subintervals. The length of each subinterval, Δx, is given by Δx = (b - a)/n.
3. Choose sample points: Choose an arbitrary sample point, denoted by xi, in each subinterval. In this case, you can choose the right endpoint of each subinterval as the sample point. The right endpoint xi is given by xi = a + iΔx, where i ranges from 1 to n.
4. Calculate the height of the rectangles: Evaluate the function f at each sample point xi to determine the height of each rectangle. In this case, the height is given by f(xi) = 3 + sin^2(xi).
5. Calculate the area of each rectangle: The area of each rectangle is given by the product of its height and width. The width is Δx, and the height is f(xi). So, the area of the i-th rectangle is given by Ai = f(xi) * Δx.
6. Sum up the areas: Sum up the areas of all rectangles from i = 1 to n to get an approximation of the area under the graph. This is expressed as the Riemann sum:
A = Σ (i = 1 to n) Ai = Σ (i = 1 to n) f(xi) * Δx
Substituting the values, we get:
A = Σ (i = 1 to n) (3 + sin^2(xi)) * Δx
Note: The notation Σ (i = 1 to n) represents the summation of all values from i = 1 to n.
7. Taking the limit: To find the exact area under the graph, we need to take the limit of the Riemann sum as the number of subintervals approaches infinity. This is denoted as the definite integral:
A = lim (n -> ∞) Σ (i = 1 to n) (3 + sin^2(xi)) * Δx
To evaluate the limit and find the exact expression for the area, further steps are needed. However, as per your request, we will stop here without evaluating the limit.
To find an expression for the area under the graph of f using the given definition, we need to consider approximating rectangles.
Based on the given definition, we can approximate the area under the curve by dividing the interval [0, π] into n subintervals.
The width of each subinterval, Δx, is given by Δx = (π - 0)/n = π/n.
Now, let's consider each subinterval and find the height of the rectangle to approximate the area under the curve.
We evaluate the function f(x) = 3 + sin^2(x) at various points within each subinterval.
For the ith subinterval, the left endpoint is xi = 0 + (i - 1)Δx, and the right endpoint is xi' = 0 + iΔx.
The height of the rectangle for the ith subinterval, hi, is given by the function evaluated at the midpoint, which is in this case, (xi + xi')/2.
So, for the ith subinterval, the height hi is equal to:
hi = f((xi + xi')/2) = f((0 + (i - 1)Δx + 0 + iΔx)/2) = f(iΔx/2) = 3 + sin^2((iπ)/2n).
Now, we can express the area of each rectangle as the product of its width and height, which gives:
Ai = Δx * hi = π/n * (3 + sin^2((iπ)/2n)).
To find the area under the graph as a limit, we sum up the areas of all the rectangles:
A = lim(n→∞) Σ(i=1 to n) Ai = lim(n→∞) Σ(i=1 to n) (π/n * (3 + sin^2((iπ)/2n))).
Note that we don't need to evaluate this limit; we simply leave it as an expression.
So, the expression for the area under the graph of f as a limit, using the given definition and endpoints, is:
A = lim(n→∞) Σ(i=1 to n) (π/n * (3 + sin^2((iπ)/2n))).