Question

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using \(n=16,32,\) and 64 subintervals. Make a conjecture about the limits of the approximations. The region bounded by the graph of \(f(x)=\left|1-x^{3}\right|\) and the \(x\) -axis on the interval [-1,2].

Short answer

#Answer# When calculating the area under the curve for an absolute value function, we handle it by considering the positive and negative parts of the function separately. It is important because the absolute value function may have distinct regions where the function is positive or negative, and integrating each region individually ensures that we account for the entire area under the curve properly.

Step-by-step solution

Define the function f(x)

The function is given by \(f(x)=\left|1-x^{3}\right|\). Since we are working with absolute values, we need to consider both the positive and the negative parts of the function separately. We will first calculate the area under the curve for the positive part of the function by finding the x-intercept, and then we will calculate the area under the curve for the negative part of the function using the same method.

Set up the integral

To find the area under the curve, we will integrate f(x) over the interval [-1, 2]. Since the function is continuous, we will first find the x-intercept (where f(x) = 0) by setting \(1-x^3=0\). We have: $$x^3=1$$ $$x=1$$ Now we have two regions to consider, [-1,1] and [1,2].

Find the integral

Since \(f(x)\) has an absolute value, we must consider the negative and positive parts separately and then add the two areas together: $$A = \int_{-1}^{1}(1-x^3)dx + \int_{1}^{2}(x^3-1)dx$$

Create the Sections

We need to approximate the area using 16, 32, and 64 subintervals. We can break the interval into three separate sets of 16, 32, and 64 subintervals each, totaling 112 subintervals from -1 to 2. We will evaluate the area under the curve for each subinterval and totalize the results.

Sum the Areas

To find the sum of the areas of all subintervals, we will consider the midpoint of each subinterval and evaluate f(x) at that point. Then, we will multiply this value by the width of the subinterval to calculate the area. We will repeat this procedure for all subintervals and sum the results to find the final estimate of the area under the curve.

Show Results in a Table

Finally, we will summarize the approximate areas calculated using n=16, 32, and 64 subintervals in a table format. | n | Area | |-------:|------------:| | 16 | Area_16 | | 32 | Area_32 | | 64 | Area_64 |

Make a Conjecture

Observing the values of the approximated areas for each n, we can make a conjecture about the limit of the approximation as n gets larger. As n increases, we can anticipate that the approximation values will get closer to the actual value of the area under the curve.

Key concepts

Riemann Sums

At the heart of calculus lies the idea of understanding and approximating areas under curves. One foundational method for doing this is using Riemann sums.

Riemann sums allow us to approximate the area under a curve by dividing the region into smaller subintervals, each taking the shape of a rectangle. The sum of the areas of these rectangles gives us a rough estimate of the total area. To perform a Riemann sum calculation, we choose a number of subintervals, say, 'n'. For each subinterval, a sample point is chosen - often the left end, the right end, or the midpoint. The function value at this sample point is then multiplied by the width of the subinterval to get the area of the rectangle.

For instance, if we're approximating the area under the function \( f(x)=|1-x^{3}| \) between \( x=-1 \) and \( x=2 \), we could divide this interval into 'n' subintervals and calculate the sum of the areas of rectangles formed at each subinterval. The narrower these subintervals (larger 'n'), the more rectangles we have, and the more accurate our approximation becomes.

When dealing with functions that include absolute values, like in our exercise, we need to take into account that the function's graph can have distinct segments where it's above and below the x-axis. This requires us to break our area approximation into parts where the function is positive and where it is negative and then add the absolute values of these areas together to ensure we have the total area.

Definite Integrals

Moving beyond approximation, we have the concept of definite integrals, which gives us the exact area under a curve within a specific interval. The definite integral is denoted as \( \int_{a}^{b} f(x) dx \), where 'a' and 'b' are the bounds of integration.

In essence, the definite integral is the limit of the Riemann sums as the number of subintervals approaches infinity. This is the precise value that the Riemann sums are approximating. For the function \( f(x)=|1-x^{3}| \), for example, we use definite integrals to calculate the exact area under the curve from \( x=-1 \) to \( x=2 \), considering the change in sign due to the absolute value.

The calculation of a definite integral involving absolute values requires us to identify where the function changes from positive to negative or vice versa since the integral takes into account the orientation of the area relative to the x-axis. This is why, in step 3 of the solution, we find the regions where the function is above and below the x-axis and integrate them separately.

Calculus Area Approximation

The process of approximating areas using calculus is an essential tool in the sciences and engineering. The method of Riemann sums can be viewed as the precursor to the precise calculations made using definite integrals.

Area approximation typically starts with simpler methods like Riemann sums to gain an initial understanding. The exercises you see in textbooks are designed to reinforce these concepts by applying them to various functions. As the number of subintervals increases (16, 32, 64, ...), the approximation generally gets better.

In our exercise, Approximation Tables often show this progressive refinement. We would see that as 'n' grows, the closer the Riemann sum gets to the exact area, which we would determine using the definite integral. This demonstrates the concept of convergence in action – with enough subintervals; our estimation converges to the true area.

In practical applications, such as evaluating the distance traveled by a vehicle when acceleration varies, calculus area approximation methods are used to gauge quantities that are otherwise challenging to measure directly. Thus, mastery of Riemann sums and definite integrals is not just academic; it has real-world implications.