Use the limit process to find the area of the region between the graph of the function and the x-axis over the indicated interval. Sketch the region. y=2x-x^(3), [1,3]
To find the area between the graph of the function y = 2x - x^3 and the x-axis over the interval [1, 3], we can use the limit process. Here's how you can do it:
1. Start by sketching the graph of the function y = 2x - x^3 within the given interval [1, 3]. This will help visualize the region whose area we want to find.
2. Divide the interval [1, 3] into n equal subintervals, each of width Δx = (b - a)/n, where a = 1 and b = 3 are the endpoints of the interval. Let's call the width of each subinterval Δx.
3. Choose a representative x-value within each subinterval. We can choose these representative x-values as the right endpoints of the subintervals. The right endpoint of the first subinterval is x₁ = 1 + Δx, the right endpoint of the second subinterval is x₂ = 1 + 2Δx, and so on, until the right endpoint of the last subinterval is xₙ = 1 + nΔx.
4. Evaluate the function y = 2x - x^3 at each representative x-value chosen in the previous step. This will give us the corresponding y-values for each subinterval.
5. The area of each small rectangle formed by one subinterval is given by the product of its width (Δx) and the height of the rectangle, which is the corresponding y-value of the function evaluated at the representative x-value.
6. Sum up the areas of all the small rectangles from the first subinterval to the last subinterval. This can be done using the summation notation: Σ[1, n] Δx * y.
7. Finally, take the limit of this sum as the number of subintervals (n) approaches infinity. This will give us the exact area of the region between the graph of the function and the x-axis over the interval [1, 3].
However, if you're looking for a quick way to find the area without going through the limit process, you can also use integral calculus. The definite integral of 2x - x^3 from x = 1 to x = 3 will give you the area of the region directly.