These are the two problems from my homework I don't get.. can you help me?
A: y=2x-x^3 [0,1] use the limit process to find the area of the
region between the graph of the function and the x-axis over the
given interval. Sketch the region
B: f(x)=4-(12/(x^2))Calculate the lower and upper sum estimates of the area under the curve over the interval [2,5] using subintervals of width ½.
Thanks!
Sure, I can help you with both of these problems. Let's start with problem A:
A: To find the area of the region between the graph of the function y = 2x - x^3 and the x-axis over the interval [0,1], we can use the limit process. Here are the steps:
1. Divide the interval [0,1] into n equal subintervals. Let the width of each subinterval be Δx.
2. Choose a point xi in each subinterval. In this case, we can choose xi as any value in the subinterval, e.g., xi = x0 + iΔx, where x0 is the starting point of the interval (0 in this case) and i varies from 0 to n-1.
3. Calculate the height of the function y = 2x - x^3 at each point xi. In this case, the height would be y = 2(xi) - (xi)^3.
4. Find the area of each rectangle formed by the width Δx and the height of the function at each point xi. The area of each rectangle is ΔA = Δx * (2xi - (xi)^3).
5. Sum up the areas of all the rectangles by adding them together. This can be calculated as the limit of the Riemann sum as n approaches infinity:
Area = lim(n→∞) ∑(i=0 to n-1) ΔA
6. Simplify the expression for the sum if possible, and evaluate it to find the area.
Now, let's move on to problem B:
B: To calculate the lower and upper sum estimates of the area under the curve of the function f(x) = 4 - (12/(x^2)) over the interval [2,5] using subintervals of width ½, we can follow these steps:
1. Divide the interval [2,5] into subintervals of equal width. In this case, the width of each subinterval is ½, so we have (5 - 2) / (1/2) = 6 subintervals.
2. Calculate the function value at the left endpoint of each subinterval. In this case, the left endpoint of each subinterval would be xi, and the function value at xi would be f(xi) = 4 - (12/(xi)^2).
3. Multiply the function value at each left endpoint by the width of each subinterval to find the area of the rectangle for that subinterval. For example, the area of the rectangle for the first subinterval would be 1/2 * (4 - (12/(2)^2)) = 1/2 * (4 - 3) = 1/2.
4. Sum up the areas of all the rectangles calculated in step 3. This will give you the lower sum estimate of the area under the curve.
5. Repeat steps 2-4, but this time calculate the function value at the right endpoint of each subinterval. This will give you the upper sum estimate of the area under the curve.
By following these steps, you should be able to calculate the lower and upper sum estimates of the area under the curve for problem B over the interval [2,5].