Question

Use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{0}^{2} \sqrt{x} d x $$

Short answer

The exact value of the integral is \( \frac{4\sqrt{2}}{3} \).

Step-by-step solution

Divide the Interval

To find the approximate value of the integral using Riemann sums and other numerical methods, divide the interval from 0 to 2 into 8 equal subintervals. The width of each subinterval is \( \Delta x = \frac{2 - 0}{8} = 0.25 \). The subintervals are \( [0, 0.25], [0.25, 0.5], \ldots, [1.75, 2] \).

Left Riemann Sum

For the left Riemann sum, use the left endpoint of each subinterval. Calculate the sum: \[ L_8 = \Delta x \left( f(0) + f(0.25) + \ldots + f(1.75) \right) \]Substitute \( f(x) = \sqrt{x} \):\[ L_8 = 0.25 \left( \sqrt{0} + \sqrt{0.25} + \sqrt{0.5} + \sqrt{0.75} + \sqrt{1} + \sqrt{1.25} + \sqrt{1.5} + \sqrt{1.75} \right) \]

Right Riemann Sum

For the right Riemann sum, use the right endpoint of each subinterval. Calculate the sum: \[ R_8 = \Delta x \left( f(0.25) + f(0.5) + \ldots + f(2) \right) \]Substitute \( f(x) = \sqrt{x} \):\[ R_8 = 0.25 \left( \sqrt{0.25} + \sqrt{0.5} + \ldots + \sqrt{2} \right) \]

Trapezoidal Rule

The trapezoidal rule averages the left and right sums: \[ T_8 = \frac{L_8 + R_8}{2} \]It accounts for the area under the curve as trapezoids instead of rectangles.

Parabolic Rule (Simpson's Rule)

Simpson's rule uses parabolas to estimate the integral. It's expressed as:\[ S_8 = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{k=1,3,5,7} f(x_k) + 2 \sum_{k=2,4,6} f(x_k) + f(x_8) \right) \]Here, \( x_k \) are the endpoints of the subintervals. Calculate the sum by substituting all terms.

Exact Value using the Second Fundamental Theorem of Calculus

To find the exact value, first determine the antiderivative of \( \sqrt{x} \), which is \( \frac{2}{3}x^{3/2} \). Apply the definite integral:\[ \int_0^2 \sqrt{x} \, dx = \left[ \frac{2}{3}x^{3/2} \right]_0^2 = \frac{2}{3}(2^{3/2}) - \frac{2}{3}(0^{3/2}) = \frac{2}{3} \times 2\sqrt{2} = \frac{4\sqrt{2}}{3} \]

Compare Approximations with Exact Value

Calculate or approximate each of the numerical methods and compare them to the exact value \( \frac{4\sqrt{2}}{3} \). This illustrates how accurate the methods are.

Key concepts

Riemann Sums

Numerical integration often uses Riemann sums to approximate definite integrals. A Riemann sum works by dividing an interval into smaller subintervals and using the sum of rectangular areas to estimate the area under a curve.
In the example of the integral from 0 to 2 of \( \sqrt{x} \), the interval is divided into 8 subintervals of equal width \( \Delta x = 0.25 \).

There are two primary types of Riemann sums:

• **Left Riemann Sum**: Uses the left endpoint of each subinterval. For example, use \( f(0), f(0.25) \), etc., within the 8 defined intervals to estimate the integral.
• **Right Riemann Sum**: Utilizes the right endpoint. Here, points like \( f(0.25), f(0.5) \), and so on, are used for calculations.

The Riemann sum provides a foundational approach to approximating the integral before exploring more precise methods.

Trapezoidal Rule

The trapezoidal rule refines the approximation by using trapezoids instead of rectangles, which helps account for the curve better. This method essentially averages out the left and right Riemann sums. This means instead of treating each section under the curve as a rectangle, we form a trapezoid, recognizing the slant of the curve.

The formula is:
\[ T_8 = \frac{L_8 + R_8}{2} \]

Here, \( L_8 \) and \( R_8 \) refer to the left and right Riemann sums, respectively, calculated for 8 subintervals.

By using the trapezoidal rule, you gain a more precise estimate for the area under \( \sqrt{x} \) over \([0, 2]\). This method offers simplicity and improved precision over basic Riemann sums.

Simpson's Rule

Simpson's Rule goes beyond rectangles and trapezoids by estimating the area under a curve using parabolic arcs. This is usually more accurate, especially for functions that are smooth and continuously differentiable.

Simpson's Rule formula for \( n = 8 \) is:

\[ S_8 = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{k=1,3,5,7} f(x_k) + 2 \sum_{k=2,4,6} f(x_k) + f(x_8) \right) \]

Simpson's Rule effectively uses every second point to form a parabola, taking into account each adjacent pair of intervals. This provides a remarkably accurate estimate for the integral of \( \sqrt{x} \) from 0 to 2.

By applying Simpson's Rule, one tends to get very close approximations to the actual area, making it a reliable choice for many calculus problems.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects the process of differentiation with integration, offering a way to evaluate definite integrals analytically rather than numerically.

Particularly, the Second Fundamental Theorem states that if \( F \) is an antiderivative of \( f \), then:

\[ \int_a^b f(x) \, dx = F(b) - F(a) \]

For the integral \( \int_0^2 \sqrt{x} \, dx \), we first find the antiderivative of \( \sqrt{x} \), which is \( \frac{2}{3}x^{3/2} \).

Then apply the definite integral formula:

\[ \left[ \frac{2}{3}x^{3/2} \right]_0^2 = \frac{4\sqrt{2}}{3} \]

This provides the exact area under \( \sqrt{x} \) from 0 to 2. Comparing this value to the approximations from Riemann sums, the trapezoidal rule, and Simpson's rule highlights the efficiency and accuracy of each method.