Questions: Approximate the area of the region between the graph of the function g(x)=100x-x^3 and the x-axis on the interval [0,10]. Use n=4 subintervals, and use the right endpoint of each subinterval when approximating the area for each subinterval. If necessary, round any intermediate calculations to no less than six decimal places and round your final answer to four decimal places.

Approximate the area of the region between the graph of the function g(x)=100x-x^3 and the x-axis on the interval [0,10]. Use n=4 subintervals, and use the right endpoint of each subinterval when approximating the area for each subinterval. If necessary, round any intermediate calculations to no less than six decimal places and round your final answer to four decimal places.

Transcript text: Approximate the area of the region between the graph of the function $g(x)=100 x-x^{3}$ and the $x$-axis on the interval $[0,10]$. Use $n=4$ subintervals, and use the right endpoint of each subinterval when approximating the area for each subinterval. If necessary, round any intermediate calculations to no less than six decimal places and round your final answer to four decimal places.

Solution

Solution Steps

To approximate the area under the curve $ g(x) = 100x - x^3 $ on the interval $[0, 10]$ using $ n = 4 $ subintervals and the right endpoint of each subinterval, we can use the Riemann sum method. The steps are as follows:

Divide the interval $[0, 10]$ into 4 equal subintervals.
Calculate the width of each subinterval.
Determine the right endpoint of each subinterval.
Evaluate the function at each right endpoint.
Multiply each function value by the width of the subinterval and sum these products to get the approximate area.

Step 1: Define the Function and Interval

We are given the function $ g(x) = 100x - x^3 $ and the interval $[0, 10]$. We will approximate the area under the curve of this function on the specified interval.

Step 2: Determine the Number of Subintervals

We will use $ n = 4 $ subintervals to divide the interval $[0, 10]$.

Step 3: Calculate the Width of Each Subinterval

The width $ \Delta x $ of each subinterval is calculated as: \[ \Delta x = \frac{b - a}{n} = \frac{10 - 0}{4} = 2.5 \]

Step 4: Identify the Right Endpoints

The right endpoints of the subintervals are:

For the first subinterval: $ x_1 = 0 + 1 \cdot 2.5 = 2.5 $
For the second subinterval: $ x_2 = 0 + 2 \cdot 2.5 = 5.0 $
For the third subinterval: $ x_3 = 0 + 3 \cdot 2.5 = 7.5 $
For the fourth subinterval: $ x_4 = 0 + 4 \cdot 2.5 = 10.0 $

Thus, the right endpoints are $ [2.5, 5.0, 7.5, 10.0] $.

Step 5: Evaluate the Function at the Right Endpoints

We calculate $ g(x) $ at each right endpoint:

$ g(2.5) = 100(2.5) - (2.5)^3 = 250 - 15.625 = 234.375 $
$ g(5.0) = 100(5.0) - (5.0)^3 = 500 - 125 = 375 $
$ g(7.5) = 100(7.5) - (7.5)^3 = 750 - 421.875 = 328.125 $
$ g(10.0) = 100(10.0) - (10.0)^3 = 1000 - 1000 = 0 $

Step 6: Calculate the Approximate Area

The approximate area $ A $ under the curve is given by the sum of the areas of the rectangles formed by the right endpoints: \[ A = \sum_{i=1}^{n} g(x_i) \Delta x = (g(2.5) + g(5.0) + g(7.5) + g(10.0)) \cdot \Delta x \] Substituting the values: \[ A = (234.375 + 375 + 328.125 + 0) \cdot 2.5 = 937.5 \cdot 2.5 = 2343.75 \]