Problem 169 - Calculus Volume 1

Question

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ \int \frac{4}{x^{2}} d x \text { over }[1,4] $$

Step-by-Step Solution

Answer

The exact area is 3, and the estimated area using \(T_{10}\) is an approximation.

Step 1: Define the Function and Interval

The function we need to find the area under is \( f(x) = \frac{4}{x^2} \) over the interval \([1, 4]\). The task is to estimate the area using Riemann sums with \(N=10\) rectangles and calculate the exact area using the Fundamental Theorem of Calculus.

Step 2: Calculate Width of Rectangles

For Riemann sums, we first find the width of each rectangle, \( \Delta x \). Divide the interval length by the number of rectangles: \[ \Delta x = \frac{4 - 1}{10} = 0.3 \]

Step 3: Compute Left-Endpoint Sum

Using the left-endpoints, calculate the sum: \[ L_{10} = \sum_{i=0}^{9} f(1 + i \cdot 0.3) \cdot 0.3 \] Compute each function value: \( f(1), f(1.3), ..., f(3.7) \) and sum them:

Step 4: Compute Right-Endpoint Sum

Using the right-endpoints, calculate the sum: \[ R_{10} = \sum_{i=1}^{10} f(1 + i \cdot 0.3) \cdot 0.3 \] Compute each function value: \( f(1.3), f(1.6), ..., f(4) \) and sum them.

Step 5: Calculate Average of Left and Right Sums

Now calculate the average of both sums: \[ T_{10} = \frac{L_{10} + R_{10}}{2} \] This will give the estimated area under the curve.

Step 6: Find the Antiderivative

To use the Fundamental Theorem, Part 2, find the antiderivative of \( f(x) = \frac{4}{x^2} \), which is \( F(x) = -\frac{4}{x} \) because the derivative of \(-\frac{4}{x}\) is \(\frac{4}{x^2}\).

Step 7: Evaluate the Definite Integral

Apply the Fundamental Theorem of Calculus to find the exact area: \[ \int_{1}^{4} \frac{4}{x^2} \, dx = F(4) - F(1) = \left(-\frac{4}{4}\right) - \left(-\frac{4}{1}\right) = -1 + 4 = 3 \]

Step 8: Conclusion

The estimated area using \( T_{10} \) is an approximation, whereas the exact area found using the Fundamental Theorem of Calculus is \(3\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects the concept of differentiation with integration, which are two main operations in calculus. This powerful theorem is all about linking the behavior of functions to the area under their curves. It comes in two parts:

• Part 1: States that if you have a function that is continuous over an interval, then it has an antiderivative, which is essentially a function whose derivative brings you back to your original function.
• Part 2: This is crucial when working with definite integrals. It tells us how to evaluate the area under a curve between two points. If you know the antiderivative of the function, you can easily compute the definite integral.

In practical terms, Part 2 allows us to find the exact area by evaluating the antiderivative at the boundaries of the interval. This greatly simplifies the process, making it a central tool for solving definite integrals effectively.

Definite Integral

A definite integral calculates the area under a curve between two points on a graph. It gives you a precise value rather than a general expression. When you see a definite integral like \( \int_{a}^{b} f(x) \, dx \), you are dealing with:

• The function \( f(x) \) whose area you want to find.
• The limits \( a \) and \( b \), which denote the interval over which you're finding this area.

Utilizing definite integrals can be seen as summing infinitely many tiny areas under the curve. Thanks to the Fundamental Theorem of Calculus, finding these areas is straightforward if you know the antiderivative. For example, in the exercise above, the definite integral \( \int_{1}^{4} \frac{4}{x^2} \, dx \) yielded an exact area of 3 after applying the theorem, illustrating the usefulness and power of this concept.

Antiderivative

Finding an antiderivative is like reversing differentiation. If you are given a function \( f(x) \), its antiderivative \( F(x) \) is a function such that the derivative of \( F(x) \) returns you to \( f(x) \). This process is often called "integration."Antiderivatives become particularly important when evaluating definite integrals, as they allow you to use the Fundamental Theorem of Calculus. For the function \( \frac{4}{x^2} \), the antiderivative is \( -\frac{4}{x} \) because if you differentiate \( -\frac{4}{x} \), you indeed get back \( \frac{4}{x^2} \). Once you have this antiderivative, you can find the definite integral over any interval by evaluating this antiderivative at the endpoints and subtracting.

Interval Estimation

Interval estimation is a statistical technique used to make an educated guess about the area under a curve. In calculus, a common method involves using Riemann sums. Instead of calculating exact values, you can compute the area using approximations. By dividing the interval into smaller sub-intervals (rectangles or trapezoids), you find the sum of all these smaller areas. This approach involves:

• Left-endpoint Riemann sums: Using the left endpoint of each sub-interval to estimate the height of rectangles.
• Right-endpoint Riemann sums: Using the right endpoint for the same purpose.

The exercise involved computing these sums to get a rough estimate and then finding the average for a better approximation. While this method isn't as precise as using the Fundamental Theorem of Calculus, it's very helpful for understanding integrals conceptually.