Problem 7 - Precalculus with Calculus Previews

Question

Approximate the area under the graph of \(f(x)=-\) \(x^{2}+5 x\) on the interval [0,5] using five subintervals of equal width and choosing: (a) \(x_{k}^{\text {as the left-hand endpoint of each }}\) subinterval, and (b) \(x_{k}^{\text {as the right-hand endpoint of each }}\) subinterval.

Step-by-Step Solution

Answer

Both approximations yield an area of 20.

Step 1: Calculate width of subinterval (\( \Delta x \))

The interval [0,5] is divided into 5 equal subintervals. Calculate the width \( \Delta x \) of each subinterval using the formula \( \Delta x = \frac{(b-a)}{n} \), where \( a = 0 \), \( b = 5 \), and \( n = 5 \). This gives \( \Delta x = \frac{5-0}{5} = 1 \).

Step 2: Identify endpoints for left-hand method (Part a)

For the left-hand approximation, use the left endpoint of each subinterval. The endpoints \( x_k \) are 0, 1, 2, 3, and 4.

Step 3: Compute area using left-hand endpoints (Part a)

Calculate the area under the curve using the left-hand endpoints: \[ \text{Area} = \sum_{k=0}^{4} f(x_k) \Delta x = \left( f(0) + f(1) + f(2) + f(3) + f(4) \right) \cdot 1\] where, \( f(x) = -x^2 + 5x \). Evaluate \( f(x) \) for each left endpoint. \( f(0) = 0 \), \( f(1) = 4 \), \( f(2) = 6 \), \( f(3) = 6 \), \( f(4) = 4 \). So, \( \text{Area} = 0 + 4 + 6 + 6 + 4 = 20 \).

Step 4: Identify endpoints for right-hand method (Part b)

For the right-hand approximation, use the right endpoint of each subinterval. The endpoints \( x_k \) are 1, 2, 3, 4, and 5.

Step 5: Compute area using right-hand endpoints (Part b)

Calculate the area under the curve using the right-hand endpoints: \[ \text{Area} = \sum_{k=0}^{4} f(x_k) \Delta x = \left( f(1) + f(2) + f(3) + f(4) + f(5) \right) \cdot 1\] Evaluate \( f(x) \) for each right endpoint. \( f(1) = 4 \), \( f(2) = 6 \), \( f(3) = 6 \), \( f(4) = 4 \), \( f(5) = 0 \). So, \( \text{Area} = 4 + 6 + 6 + 4 + 0 = 20 \).

Key Concepts

Left Endpoint Approximation

Left Endpoint Approximation is a numerical method used to estimate the area under a curve from a function, using statistical partitions. It's crucial when you're dealing with complex functions that are difficult to integrate analytically.
In this type of approximation, you divide the interval of interest into smaller subintervals that are of equal width. Then, for each subinterval, you take the function's value at the left endpoint. This value represents the height of the rectangle.

• Divide the interval \[a, b\] into \['n'\] equal parts. The width of each subinterval is \[ \Delta x = \frac{b-a}{n} \].
• Identify the left endpoints for the subintervals: these form the values where you calculate your function \(f(x)\).
• Sum up the areas of the rectangles by multiplying \(f(x)\) at each left endpoint by \(\Delta x\).

For instance, consider the polynomial function \(f(x) = -x^2 + 5x\) over the interval \[0, 5\]. By using five subintervals and computing at points 0, 1, 2, 3, and 4, you sum the areas and find that they approximately total to 20.

Right Endpoint Approximation

Right Endpoint Approximation is another useful technique to find the area under a curve, particularly in approximating integrals for complex functions.
Unlike the left endpoint method, you calculate the height of the approximating rectangles using the right endpoint of each subinterval.
Here's how it works:

• Again, divide the interval into equal parts, determining the width \( \Delta x = \frac{b-a}{n} \).
• Instead, use the right endpoints of each subinterval. These are the points right before the next subinterval starts.
• Apply these right endpoints to your function \(f(x)\) to get the height of each rectangle, and multiply each by \(\Delta x\) to find the area.
• Add up these areas to get your approximation.

Using the function \(f(x) = -x^2 + 5x\), and for the interval \[0, 5\], determine the right endpoints: 1, 2, 3, 4, and 5. As you compute and sum these endpoints' contributions, the total estimated area remains 20, matching the result from the left endpoint method.

Polynomial Functions

Polynomial functions are fascinating and play a fundamental role in calculus and many areas of mathematics. A polynomial function is expressed as the sum of terms, where each term includes a coefficient and a variable raised to a non-negative integer exponent.
The general form of a polynomial function is:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]

• The highest exponent of the variable 'x' in the function is called the degree of the polynomial.
• Polynomial functions can be linear, quadratic, cubic, etc. based on their degree.

In exercises involving Riemann sums and endpoint approximations, polynomial functions like \( f(x) = -x^2 + 5x \) are often used. This quadratic polynomial, for instance, highlights the curve's behavior and allows you to practice using these numerical approximation methods. They give insight into how the algebraic expression can describe smooth and continuous phenomena over an interval.