This is a question from my textbook that does't have a solution and quite frankly I have no idea what to do. Any tips would be greatly appreciated.
Given the function f defined by f(x) = 9 - x^2. Find the surface area bounded by the curve y = f(x), the x axis and the lines x = 2 and x = 0.
(a) Find the Riemann Sn sum algebraically, obtained by cutting the surface in n intervals of equal width and considering circumscribed rectangles
(b)Evaluate the surface area by finding lim n->infinite Sn
Thank you
for your first part of the question,
area = ∫(9-x^2) dx from x=0 to x=2
= 9x- (1/3)x^3 | from 0 to 2
= 18 - (1/3)(8) - 0
= 18 - 8/3
= 46/3 units^2
The last time I taught Riemann's sums was about 50 years ago, so I will not attempt that part.
Thanks Reiny :)
To find the surface area bounded by the curve y = f(x), the x-axis, and the lines x = 2 and x = 0, we can use Riemann Sums and take the limit as the number of intervals approaches infinity.
(a) Riemann Sums with circumscribed rectangles involves partitioning the interval [0, 2] into n equal subintervals, and then approximating the surface area by summing the areas of rectangles that are drawn tangent to the curve.
Here are the steps to find the Riemann Sn sum algebraically:
1. Partition the interval [0, 2] into n equal subintervals. Each subinterval will have a width of Δx = (2 - 0)/n = 2/n.
2. Choose a representative point in each subinterval, denoted by xi*. The representative point can be any point in the subinterval, such as the right or left endpoint, or the midpoint. For simplicity, let's choose the midpoint of each subinterval.
3. Compute the height of each rectangle by substituting the representative point xi* into the function f(x). So, the height of each rectangle will be f(xi*).
4. Calculate the width and the height of each rectangle and multiply them together to find the area of each rectangle. The area of the i-th rectangle will be Ai = f(xi*) * Δx.
5. Sum up the areas of all the rectangles to find the Riemann Sn sum. The Riemann Sn sum is given by Sn = Σ Ai = Σ f(xi*) * Δx, ranging from i = 1 to n.
(b) To evaluate the surface area by finding the limit as n approaches infinity (lim n→∞ Sn), you need to take the sum of rectangles for an increasingly large number of intervals. Essentially, you need to show that as n increases without bound, the Riemann Sn sum approaches a certain value.
To find lim n→∞ Sn, you can rearrange the Riemann Sn sum and apply the limit operation. The key is to recognize that the Riemann Sn sum is actually a Riemann integral. The limit of the Riemann sum as n approaches infinity is equivalent to the definite integral of the function f(x) over the interval [0, 2]. So, you can rewrite the limit as:
lim n→∞ Sn = ∫[0, 2] f(x) dx.
To find the exact value of the surface area, you need to evaluate this definite integral.
Please note that in this case, the definite integral can be evaluated directly. The surface area can be obtained by integrating the function f(x) = 9 - x^2 from x = 0 to x = 2.