Questions: Approximate the area under the curve y=x^3 from x=2 to x=5 using a Right Endpoint approximation with 6 subdivisions.

Approximate the area under the curve y=x^3 from x=2 to x=5 using a Right Endpoint approximation with 6 subdivisions.

Transcript text: Approximate the area under the curve $y=x^{3}$ from $x=2$ to $x=5$ using a Right Endpoint approximation with 6 subdivisions.

Solution

Solution Steps

Step 1: Calculate the width of each subdivision

The width of each subdivision, \(\Delta x\), is calculated as \(\Delta x = \frac{b-a}{n} = \frac{5-2}{6} = 0.5\).

Step 2: Determine the x-coordinates of the right endpoints

For each subdivision \(i\), the x-coordinate of the right endpoint, \(x_i\), is calculated as \(x_i = a + i\Delta x\).

Step 3: Evaluate the function at each right endpoint

The function \(f(x)\) is evaluated at each right endpoint \(x_i\) to find \(f(x_i)\).

Step 4: Calculate the area of each rectangle

For each subdivision, the area is approximately \(f(x_i) \cdot \Delta x\), where \(f(x_i)\) is the function value at the right endpoint of the subdivision.

Step 5: Sum the areas of all rectangles

The approximate total area under the curve is \(\sum_{i=1}^{n} f(x_i) \cdot \Delta x = 182.81\).

Final Answer: The approximate area under the curve from \(x = 2\) to \(x = 5\) using a Right Endpoint approximation with \(n = 6\) subdivisions is 182.81.

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