(a) Estimate the area under the graph of f(x) = 3sqrtx from x = 0 to x = 4 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.)
R4 = ?
To estimate the area under the graph of f(x) = 3√x from x = 0 to x = 4 using four right endpoints, we can divide the interval [0, 4] into four equal subintervals.
First, let's calculate the width of each subinterval:
Width of each subinterval = (b - a) / n
where b is the upper limit of integration (4 in this case), a is the lower limit of integration (0 in this case), and n is the number of subintervals (4 in this case).
Width of each subinterval = (4 - 0) / 4 = 1
Next, we need to determine the right endpoint for each subinterval:
Right endpoint = a + (width of each subinterval)
where a is the lower limit of integration and width of each subinterval is calculated above.
Right endpoint for the first subinterval = 0 + 1 = 1
Right endpoint for the second subinterval = 1 + 1 = 2
Right endpoint for the third subinterval = 2 + 1 = 3
Right endpoint for the fourth subinterval = 3 + 1 = 4
Now, we can calculate the area of each rectangle formed by the right endpoints:
Area of each rectangle = f(right endpoint) * (width of each subinterval)
For f(x) = 3√x, the height of each rectangle will be f(right endpoint) = 3√(right endpoint).
Area of the first rectangle = 3√1 * 1 = 3 * 1 = 3
Area of the second rectangle = 3√2 * 1 = 3 * 1.4142 = 4.2426
Area of the third rectangle = 3√3 * 1 = 3 * 1.7321 = 5.1962
Area of the fourth rectangle = 3√4 * 1 = 3 * 2 = 6
Finally, to estimate the area under the graph, we sum up the areas of all four rectangles:
R4 = Area of the first rectangle + Area of the second rectangle + Area of the third rectangle + Area of the fourth rectangle
R4 = 3 + 4.2426 + 5.1962 + 6
R4 ≈ 18.4388
Therefore, the estimated area under the graph of f(x) = 3√x from x = 0 to x = 4 using four approximating rectangles and right endpoints is approximately 18.4388 square units.