Problem 11 - Precalculus: Mathematical for Calculus

Question

Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. $$y=3 x, \quad 0 \leq x \leq 5$$

Step-by-Step Solution

Answer

The area under the curve \(y = 3x\) from \(x=0\) to \(x=5\) is \(37.5\).

Step 1: Divide the Interval

To find the area under the curve using the definition of area as a limit, first divide the interval \([0, 5]\) into \(n\) subintervals of equal width. The width of each subinterval is \(\Delta x = \frac{5-0}{n} = \frac{5}{n}\).

Step 2: Choose Sample Points

Next, choose a sample point within each subinterval. For simplicity, choose the right endpoint of each subinterval \(x_i = \frac{5i}{n}\) where \(i = 1, 2, ..., n\).

Step 3: Calculate the Function Value at Sample Points

Evaluate the function \(y = 3x\) at each sample point \(x_i\). Thus, \(f(x_i) = 3 \left(\frac{5i}{n}\right) = \frac{15i}{n}\).

Step 4: Form the Riemann Sum

The approximate area under the curve is the sum of the areas of the rectangles formed:\[ S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \frac{15i}{n} \cdot \frac{5}{n} = \sum_{i=1}^{n} \frac{75i}{n^2} \]\.

Step 5: Simplify the Sum

Using the formula \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \), simplify the sum:\[ S_n = \frac{75}{n^2} \sum_{i=1}^{n} i = \frac{75}{n^2} \cdot \frac{n(n+1)}{2} = \frac{75(n+1)}{2n} \].

Step 6: Take the Limit as n Approaches Infinity

To find the exact area under the curve, take the limit as \(n\to\infty\):\[ \lim_{n \to \infty} \frac{75(n+1)}{2n} = \lim_{n \to \infty} \left(\frac{75}{2} + \frac{75}{2n}\right) = \frac{75}{2} \].

Step 7: Verify with Geometry

Sketch the region and note that it forms a right triangle with base 5 and height 15 (since \(y = 3 \times 5 = 15\)). The area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 15 = \frac{75}{2} = 37.5\).

Key Concepts

Riemann Sum

The concept of a Riemann sum is fundamental in calculus for approximating the area under a curve. It involves breaking the area into small rectangles and calculating the sum of their areas to approximate the total area. To start, divide the interval over which you want to find the area. For the function \(y = 3x\), the interval is \([0, 5]\).
\[\]

• Divide this interval into \(n\) equal parts, each of width \(\Delta x = \frac{5}{n}\).
• Choose sample points within each subinterval; here, we use the right endpoint, \(x_i = \frac{5i}{n}\).
• Calculate the function value at these points: \(f(x_i) = 3 \left(\frac{5i}{n}\right) = \frac{15i}{n}\).

The Riemann sum becomes the sum of the rectangles' areas, formulated as:\[ S_n = \sum_{i=1}^{n} \frac{15i}{n} \cdot \frac{5}{n} \ = \sum_{i=1}^{n} \frac{75i}{n^2} \].
As \(n\) increases, the approximation gets closer to the true area.

Limit Definition of Area

Understanding the limit definition of area is key in solving problems involving continuous regions. The Riemann sum offers an approximation; however, for the exact area, we need to take the limit of this sum as the number of rectangles \(n\) approaches infinity.
\[\]

• Consider the formula obtained from simplifying the sum: \(S_n = \frac{75(n+1)}{2n}\).
• Next, compute the limit: \[ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left(\frac{75}{2} + \frac{75}{2n}\right) \].
• As \(n\) approaches infinity, the term \(\frac{75}{2n}\) approaches zero, simplifying the expression to \(\frac{75}{2}\).

This result gives the exact area under the curve \(y = 3x\) from \(x = 0\) to \(x = 5\).

Functions and Graphs

To visualize the process of finding the area, drawing the function's graph can be immensely helpful. The function \(y = 3x\) is a straight line passing through the origin with a slope of 3. Let's sketch this graph and interpret it:

• The line passes through points like (0,0), (1,3), and (5,15).
• The area under this line from \(x = 0\) to \(x = 5\) forms a triangle.
• The base of the triangle runs along the x-axis from 0 to 5, and the height, corresponding to the line at \(x = 5\), is 15.

The area of this triangle is calculated using:\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 15 = \frac{75}{2} = 37.5 \].This geometric confirmation assures that our calculus-based area calculation is correct.