Find the area of the enclosed by the y-axis and the curves y=x^2 and
y=(x^2+x+1)*e^(-x).
...I'm supposed to use tabular method to find the area.
But I'm not sure where to start with this question. I drew the graph for y=x^2, but that's about it. Help!
thanks
To find the area enclosed by the y-axis and the given curves using the tabular method, you need to follow these steps:
1. Start by setting the two equations for the curves equal to each other: y = x^2 and y = (x^2+x+1)*e^(-x).
2. Simplify the equation for the curve to be equal to each other by expanding the second equation. It becomes: y = x^2 + xe^(-x) + e^(-x).
3. Set up a table with three columns: x, y1, and y2. In the x column, choose several x-values such that both curves are defined and intersect.
4. In the y1 column, calculate the y-values for the first curve, which is y = x^2. For each x-value, simply square the x-value to get the y-value.
5. In the y2 column, calculate the y-values for the second curve, which is y = x^2 + xe^(-x) + e^(-x). For each x-value, substitute it into the equation to get the corresponding y-value.
6. Now, calculate the differences between the y-values in the y2 column and the corresponding y-values in the y1 column. Record these differences in another column, which we'll call Δy.
7. The area enclosed between the two curves can be approximated by summing up the absolute values of the differences (∑|Δy|) obtained in the previous step.
8. Finally, multiply ∑|Δy| by the width of each x-interval to obtain the estimated area enclosed.
Remember that the more partitions you make in the table and the smaller the width of each interval, the more accurate your estimation of the area will be.