Find the area of the region enclosed by the curve y=x cosx...

Find the area of the region enclosed by the curve $y=x \cos x$ and the $x$ -axis (see the accompanying figure) for a. $\pi / 2 \leq x \leq 3 \pi / 2 $ b. $3 \pi / 2 \leq x \leq 5 \pi / 2$ c. $5 \pi / 2 \leq x \leq 7 \pi / 2$ d. What pattern do you see? What is the area between the curve and the $x$-axis for $$\left(\frac{2 n-1}{2}\right) \pi \leq x \leq\left(\frac{2 n+1}{2}\right) \pi$$ $n$ an arbitrary positive integer? Give reasons for your answer.

Find the area of the region enclosed by the curve $y=x \cos x$ and the $x$ -axis (see the accompanying figure) for
a. $\pi / 2 \leq x \leq 3 \pi / 2 $
b. $3 \pi / 2 \leq x \leq 5 \pi / 2$
c. $5 \pi / 2 \leq x \leq 7 \pi / 2$
d. What pattern do you see? What is the area between the curve and the $x$-axis for
$$\left(\frac{2 n-1}{2}\right) \pi \leq x \leq\left(\frac{2 n+1}{2}\right) \pi$$
$n$ an arbitrary positive integer? Give reasons for your answer.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, providing a systematic method to evaluate definite integrals using antiderivatives. It serves as the foundation for calculating the exact area under a curve by transforming the integration problem into the computation of antiderivative values at the endpoints of an interval.

Pattern Recognition in Sequences of Integrals

Recognizing patterns in a sequence of integrals involves analyzing the behavior of integrals computed over adjacent or repeating intervals. This insight can lead to formulating general expressions or formulas that apply to similar problems, thereby simplifying the process of finding areas in regularly repeating contexts.

Trigonometric Functions and Periodicity

Trigonometric functions, such as cosine, exhibit periodic behavior, meaning their outputs repeat at regular intervals. This periodicity is key when analyzing patterns in integrals computed over different segments of a cycle, making it possible to generalize results over successive intervals.

Area between a Curve and the X-Axis

When computing the area between a curve and the x-axis, it is important to consider the absolute value of the function to ensure that areas below the axis (which would yield negative values under straightforward integration) are counted as positive. This concept is essential in accurately determining the total area enclosed by the curve.

Definite Integration

Definite integration is a core concept in calculus that calculates the net accumulation of quantities, such as area under a curve, over a specific interval. It uses the antiderivative of a function to evaluate the integral across given bounds, thereby determining areas, accumulated change, or total quantities in various applications.

Transcript

00:01 All right, for problem 32, we're given a curve that we need to find some areas of between the x -axis, right, x -cosine x.

00:09 The first one being from pi over two to three pi over two.

00:14 So we can see that it's going to be, the area's going to be negative, so areas can only be positive.

00:18 So we need to take the negative integral from pi over two to three pi over two of x cosine x.

00:28 So the x.

00:30 In order to do this, we're going to have to use integration by parts.

00:33 Let's use u equals x, du, will equal d x then, and then dv will equal cosine x dx, which then makes v equal to sine x.

00:47 So now, if we piece this together, that will give us negative times x sine x minus the integral from sine x of sine x, dx, right? and all of this is going to be with the same limits of integration.

01:13 So if we go ahead and integrate this really quickly, that will give us negative x, sine x, right? the integral of sine x would be negative cosine x.

01:23 So that will cancel out here, then come back there.

01:26 So minus cosine x.

01:30 There we go.

01:31 So now we can apply our limits of integration of pi over two and three pi over two.

01:38 And that will give us.

01:39 So when we get 3 pi over 2, we get negative 3 pi over 2 times the sign of x.

01:45 So the sign of 3 pi over 2 is just negative 1, which means that this will then become positive.

01:53 So we get positive 3 pi over 2, and then minus cosine of 3 pi over 2, minus cosine of 3 pi over 2, and the cosine of 3 pi over 2.

02:07 And the cosine of 3 pi over 2 is just 0.

02:09 So that will just go away.

02:12 And then all of that will be minus.

02:17 We plug in pi over two for this.