Problem 23

Question

23\. Find the exact area under the cosine curve $y=\cos x$ from $x=0$ to $x=b,$ where 0$\leqslant b \leqslant \pi / 2 .$ (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $b=\pi / 2 ?$

Step-by-Step Solution

Verified

Answer

The area under the cosine curve from $ x=0 $ to $ x=b $ is $ \sin b $, and for $ b=\pi/2 $, the area is 1.

1Step 1: Understand the Problem

We are asked to find the area under the curve of $ y = \cos x $ from $ x = 0 $ to $ x = b $. This means we need to calculate the definite integral of the cosine function over this interval.

2Step 2: Set Up the Integral

To find the area under the curve $ y = \cos x $ from $ x = 0 $ to $ x = b $, we need to set up the definite integral as follows: \[ A = \int_{0}^{b} \cos x \, dx \].

3Step 3: Compute the Integral

The integral of $ \cos x $ is $ \sin x $, so we need to evaluate $ \int \cos x \, dx = \sin x + C $. The definite integral from $ x = 0 $ to $ x = b $ is: \[ A = \left[ \sin x \right]_{0}^{b} = \sin b - \sin 0 \].

4Step 4: Calculate the Exact Value

Substitute $ \sin 0 = 0 $ into the expression obtained: $ A = \sin b - 0 = \sin b $. So, the area under the curve from $ x = 0 $ to $ x = b $ is $ \sin b $.

5Step 5: Evaluate for Specific Value of $ b $

Specifically, we want to find the area when $ b = \pi/2 $. Hence, we evaluate $ \sin \left( \pi/2 \right) = 1 $, which gives us: \[ A = 1 \].

Key Concepts

Cosine FunctionArea Under CurveCalculus Problem SolvingComputer Algebra System

Cosine Function

The function we're working with in this exercise, the cosine function, is one of the fundamental trigonometric functions. It is typically denoted as $ y = \cos x $. The cosine function is crucial in mathematics and physics because of its properties in describing oscillations and wave patterns.

**Periodicity:** The cosine function has a period of $ 2\pi $, which means it repeats its values every $ 2\pi $ units along the x-axis.
**Symmetry:** The function is even, displaying symmetry about the y-axis.
**Range:** The cosine function has a range between -1 and 1, meaning it never exceeds these values.

In the context of this exercise, we specifically consider the cosine function over the interval from $ x=0 $ to $ x=b $, where $ b $ is between 0 and $ \pi/2 $. Understanding the behavior of the cosine function on this interval helps us calculate the area under its curve using calculus.

Area Under Curve

Finding the area under a curve is a common application of integrals in calculus. In our exercise, we focus on finding the area under the cosine curve from $ x=0 $ to $ x=b $ using a definite integral.

**Definite Integrals:** These are used to calculate the net area under a curve, delivering a numerical value that represents this area.
**Cosine Curve Specific:** For the function $ y = \cos x $, the area under the curve to the x-axis for a range $ [0, b] $ is obtained through the integral $ \int_{0}^{b} \cos x \, dx $.

The result of this integral represents the exact area between the curve and the x-axis. This can be visualized as the "shaded" region on a graph. In this exercise, understanding how to set up and calculate the definite integral is key to determining this area.

Calculus Problem Solving

Solving calculus problems often involves a step-by-step approach to isolate and tackle each part of the problem. In this exercise, we are tasked with finding the area under the cosine curve.

**Step 1: Problem Understanding:** Clearly define what is being solved – the area under the curve from $ x=0 $ to $ x=b $.
**Step 2: Set Up the Integral:** Identify the integral needed, in this case, $ \int_{0}^{b} \cos x \, dx $.
**Step 3: Compute the Integral:** Use the antiderivative of the function, which is $ \sin x $.
**Step 4: Evaluate the Integral:** Substitute limits into the antiderivative to find the area, $ \sin b - \sin 0 $.
**Step 5: Substitute Specific Values:** Particularly for this exercise, when $ b = \pi/2 $, the area is evaluated as 1.

Following these steps ensures a clear path to the solution, enhancing both understanding and accuracy in calculus problem solving.

Computer Algebra System

A Computer Algebra System (CAS) is a software tool designed to facilitate symbolic mathematics operations such as algebraic manipulations, calculus computations, and more. Using a CAS can significantly ease the process of solving complex mathematical problems.

**Symbolic Computation:** CAS can manipulate and solve equations symbolically, providing exact solutions rather than numerical approximations.
**Integration Simplification:** For this exercise, a CAS can automatically compute the definite integral $ \int_{0}^{b} \cos x \, dx $, streamlining the process and reducing the potential for manual errors.
**Instant Evaluation:** Once the integral is formulated, a CAS can quickly substitute values to output specific results, like when $ b = \pi/2 $.

Using a CAS not only aids in solving problems with precision but also helps students visualize mathematical concepts, specifically when graphically representing functions and their integrals. This makes it a powerful tool in both educational and professional settings.

Problem 22

Problem 23

Other exercises in this chapter

Problem 22

Evaluate the indefinite integral. $\int \frac{\tan ^{-1} x}{1+x^{2}} d x$

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Problem 22

First make a substitution and then use integration by parts to evaluate the integral. $\int t^{3} e^{-t^{2}} d t$

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Problem 23

$$\begin{array}{c}{\text { Suppose that } F, G, \text { and } Q \text { are polynomials and }} \\ {\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}}\end{array}$$ $$\begin{ar

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Problem 23

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are

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