A region is bounded by the function y=2x^2+3 and the x-axis over the interval(0,2). Sketch the graph of the bounded region. Use the limit process to find the area of the bounded region. Explain the step in this limit process. Please explain all procedured using correct mathematical termminology.
Dr. Simmons
To sketch the graph of the bounded region, we first need to plot the function y = 2x^2 + 3 over the specified interval (0,2). The graph of the function is a parabola that opens upwards.
To find the area of the bounded region using the limit process, we can break it down into smaller rectangles and calculate their individual areas. As the width of these rectangles approaches zero, the sum of their areas will converge to the desired area of the bounded region.
Here are the steps to calculate the area using the limit process:
1. Divide the interval (0,2) into n equal parts. Each part will have a width of Δx = (2-0)/n = 2/n.
2. Choose any point within each part, denoted as xi. One common choice is the right endpoint of each interval, so xi = i(2/n), where i ranges from 1 to n.
3. Calculate the height of each rectangle by plugging the xi values into the function y = 2x^2 + 3. The height of each rectangle will be f(xi) = 2(xi)^2 + 3.
4. The area of each rectangle is equal to the width multiplied by the height. So, the area of each rectangle, Ai, is Δx × f(xi) = (2/n) × [2(i(2/n))^2 + 3].
5. The total area, A, of the bounded region is the sum of all the individual rectangle areas. Therefore, A = ∑ Ai = ∑ [(2/n) × [2(i(2/n))^2 + 3]].
6. Simplify the expression by expanding and collecting like terms. This will yield a summation formula that only involves n.
7. Evaluate the limit as n approaches infinity (∞) of the summation formula calculated in the previous step. This limit process ensures that the width of the rectangles approaches zero, giving us an accurate estimate of the area.
Once you have determined the limit, you will have found the area of the bounded region using the limit process.