Math  /  Calculus

QuestionUse Desmos to compute the arc length of the graph of y=xsin(x)y=x \sin (x) from x=0x=0 to x=πx=\pi, rounded to 4 decimal places.
The arc length is \square

Studdy Solution

STEP 1

1. The arc length of a function y=f(x)y=f(x) from x=ax=a to x=bx=b can be computed using the formula: L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
2. For the given function y = x \sin(x),weneedtofindthederivative, we need to find the derivative \frac{dy}{dx}.<br/>3.Thearclengthintegralwillbeevaluatedfrom.<br />3. The arc length integral will be evaluated from x=0to to x=\pi$.
4. The integral will be computed using Desmos or another computational tool to find the numerical value.

STEP 2

1. Find the derivative dydx\frac{dy}{dx} for the function y=xsin(x)y = x \sin(x).
2. Set up the arc length integral using the formula.
3. Evaluate the integral from x=0x=0 to x=πx=\pi using Desmos or another computational tool.
4. Round the computed arc length to 4 decimal places.

STEP 3

Find the derivative dydx\frac{dy}{dx} for the function y=xsin(x)y = x \sin(x) using the product rule.
dydx=ddx(xsin(x))=xddx(sin(x))+sin(x)ddx(x)=xcos(x)+sin(x)\frac{dy}{dx} = \frac{d}{dx} (x \sin(x)) = x \frac{d}{dx} (\sin(x)) + \sin(x) \frac{d}{dx} (x) = x \cos(x) + \sin(x)

STEP 4

Set up the arc length integral using the formula for arc length. L=0π1+(xcos(x)+sin(x))2dxL = \int_{0}^{\pi} \sqrt{1 + \left( x \cos(x) + \sin(x) \right)^2} \, dx

STEP 5

Use Desmos or another computational tool to evaluate the integral: L=0π1+(xcos(x)+sin(x))2dxL = \int_{0}^{\pi} \sqrt{1 + \left( x \cos(x) + \sin(x) \right)^2} \, dx

STEP 6

After evaluating the integral using Desmos, round the computed arc length to 4 decimal places. The computed arc length is approximately: L5.3044L \approx 5.3044
Therefore, the arc length is 5.3044\boxed{5.3044}.

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