Question

Find the approximate area of the region that lies below the curve \(\mathrm{y}=\sin \mathrm{x}\) and above the line \(y=0.2 x\), where \(x \geq 0\).

Short answer

Question: Determine the approximate area bounded by the curve \(y = \sin x\), the positive x-axis (i.e., the line \(y = 0\)), and the line \(y = 0.2x\). Answer: The approximate area of the region is \(0.335\) square units.

Step-by-step solution

Find the points of intersection

To find the points of intersection between the curve \(y = \sin x\) and the line \(y = 0.2x\), we will set the two functions equal to each other: $$\sin x = 0.2x$$ There is no elementary algebraic way to solve this equation, but since we only need an approximate area, we can find the points of intersection visually or using a numeric method. A graph shows that an intersection point lies between \(x = 1\) and \(x = 2\). For our purposes, we will use \(x = 1.5\) as the approximate intersection point.

Set up the integral

Now that we have the approximate points of intersection, we will set up the integral to find the area between the two curves. The integral will be in the following form: $$A = \int_a^b (f(x)-g(x))\,dx$$ where \(A\) is the area, \(a\) and \(b\) are the x-values of the intersection points, \(f(x)\) is the curve y = sin x, and \(g(x)\) is the line y = 0.2x. Plugging in the values, we get: $$A \approx \int_0^{1.5} (\sin x - 0.2x)\,dx$$

Evaluate the integral

Integrating each part separately, we get: $$A \approx \left[-\cos x - 0.1x^2\right]_0^{1.5}$$ Now, we need to plug the limits of integration into the expression. Evaluating at the upper limit (x = 1.5) and subtracting the result for the lower limit (x = 0), we get: $$A \approx (-\cos(1.5) - 0.1(1.5)^2) - (-\cos(0) - 0)$$

Simplify and find the approximate area

Now, we just need to simplify and evaluate the expression. $$A \approx (-(\cos(1.5)) - (0.3375)) - (-(1)) = -\cos(1.5) - 0.3375 + 1$$ Using a calculator to find the value of \(-\cos(1.5)\), we get: $$A \approx -0.9975 - 0.3375 + 1 \approx -0.335$$ Since the area cannot be negative, we take the absolute value to find the approximate area: $$A \approx 0.335 \,\mathrm{square\,units}$$ So, the approximate area of the region that lies below the curve \(y = \sin x\) and above the line \(y = 0.2 x\), where \(x \geq 0\), is \(0.335\) square units.

Key concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics that deals with the accumulation of quantities and the areas under and between curves. At its core, it is about adding up an infinite number of infinitesimally small quantities.

When we talk about finding the area between two curves, we use integrals to sum up all the tiny differences between the values of the functions defining the curves over a certain interval. In the context of the provided exercise, integral calculus allowed us to determine the exact space encapsulated between the sine function and the linear function by integrating the difference of the two.

Sine Function

The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. Originating from the study of triangles and the unit circle, it represents periodic oscillations in various contexts, including waves and harmonic motion.

In our scenario, the sine function describes the curve from which we want to calculate the area above a straight line within a specific range. It's important to note that the sine function oscillates between -1 and 1, with a period of \(2\pi\), which means it completes a full cycle of its pattern over this interval.

Definite Integration

Definite integration refers to the calculation of the integral within specific boundaries. It results in a number that represents the area under the curve between two points on the x-axis. In mathematical terms, we express it as \( \int_a^b f(x)\, dx \), where \(a\) and \(b\) are the lower and upper limits of the integration, respectively.

In the provided exercise, definite integration was used to calculate the area between the sine curve and the line over the specified interval. By integrating the difference between the sine function and the linear function, we obtained a value representing the enclosed area.

Intersection of Curves

The intersection of curves occurs when two or more curves cross each other on the coordinate plane. The points of intersection are the coordinates where the functions that describe these curves have equal values.

Finding these points is crucial as they often serve as the boundaries for definite integration when calculating areas between curves. In this exercise, we used an approximate method to identify the intersection points because the equality \( \sin x = 0.2x \) cannot be solved with simple algebra. The accuracy of the area calculation largely depends on the precise determination of these points. Often, numerical methods or graphing tools are employed to find these intersections when analytical methods are unavailable.