Given the graph of f(x) = x sinx, 0 \leq x \leq 2pi assuming that a quantity y changes at a rate...

Question:

Given the graph of {eq}f(x) = x sinx {/eq}, {eq}0 \leq x \leq 2\pi {/eq} assuming that a quantity y changes at a rate of {eq}y' = x sinx {/eq}, find by how much it will increase or decrease over {eq}\frac{3\pi}2 \leq x \leq 2\pi {/eq}.

Definite Integrals:

Recall that a definite integral can be interpreted geometrically as the area beneath a curve {eq}f (x) {/eq} and above the {eq}x {/eq}-axis. Let's think about why this is true, using motion as our guide. When on a trip and traveling at a constant speed, we know we can find the total distance we've traveled by multiplying our speed by the amount of time we've been traveling. We can always think of a product as a rectangle in the pane; if we are traveling at 60 mph and we've been driving for a half hour, we have gone {eq}60 \cdot \frac12 = 30 {/eq} mi. This is a rectangle with height 60 and length {eq}\frac12 {/eq} with lower left corner at the origin. The point is that all products of two numbers can be thought of this way. The definite integral is just a general extension of this so that instead of rectangles, the geometric shapes can be anything.

Answer and Explanation:

We have a function that we know is the derivative, i.e. rate of change, of another function:

{eq}\begin{align*} f (x) = y' = x \sin x \end{align*} {/eq}

and we want to find the total change in the function {eq}y {/eq}, so what we want to do is multiply this rate, which tells us how {eq}y {/eq} changes with respect to {eq}x {/eq}, by the total change in {eq}x {/eq}. We want the area of the region between the curve and the {eq}x {/eq}-axis, i.e. we want to evaluate a definite integral over a specified interval:

{eq}\begin{align*} y &= \int_{3\pi/2}^{2\pi} x \sin x \ dx \\ &= \left [ \sin x- x\cos x \right ]_{3\pi/2}^{2\pi} \\ &= \sin 2\pi - 2\pi \cos 2\pi - \sin \frac{3\pi}2 + \frac{3\pi}2 \cos \frac{3\pi}2 \\ &= 0 - 2\pi(1) -(-1) - 0 \\ &= 1-2\pi \\ &\approx -5.2832 \end{align*} {/eq}

This is the change in {eq}y {/eq} we seek. One way to think about this is that {eq}y {/eq} is the signed area between the curve and the {eq}x {/eq}-axis, so the change in {eq}y {/eq} given some change in {eq}x {/eq} (our interval) is equal to the area between the curve and the {eq}x {/eq}-axis on this interval. The fact that we have a negative number means that on the interval the curve is mostly below the {eq}x {/eq}-axis.


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Definite Integrals: Definition

from Math 104: Calculus

Chapter 12 / Lesson 6
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