Final answer:
The total area between the x-axis and the graph of y = x1/5 - x from x = -1 to x = 3 is found by calculating the definite integral of the function over that interval and summing the areas above and below the x-axis.
Step-by-step explanation:
The question asks to find the total area between the x-axis and the graph of the function y = x1/5 - x for the interval -1≤x≤3. To solve this, we need to calculate the definite integral of the function from x = -1 to x = 3. This involves finding the integral of x1/5 and x separately, and then evaluating this integral from -1 to 3.
Step-by-step explanation:
- Set up the integral: ∫(x1/5 - x) dx from x = -1 to x = 3.
- Calculate the antiderivative of x1/5 which is ⅖ x6/5, and the antiderivative of x which is ½ x2.
- Substitute the upper and lower limits into the antiderivative to evaluate the integral.
- Add together the areas computed above to find the total area, accounting for areas above and below the x-axis as positive and negative respectively.