Math  /  Calculus

QuestionEvaluate: (7yesin(x))dx+[15xsin(y3+8y)]dy\oint\left(7 y-e^{\sin (x)}\right) d x+\left[15 x-\sin \left(y^{3}+8 y\right)\right] d y using one method. CC is the boundary of the graph of a circle of radius 3 .

Studdy Solution
Set up and evaluate the double integral over the region R R , which is the interior of the circle of radius 3 centered at the origin:
The area dA dA in polar coordinates is rdrdθ r \, dr \, d\theta . The limits for r r are from 0 to 3, and for θ \theta are from 0 to 2π 2\pi .
R8dA=02π038rdrdθ\iint_R 8 \, dA = \int_0^{2\pi} \int_0^3 8 \cdot r \, dr \, d\theta
Evaluate the inner integral:
038rdr=8[r22]03=892=36\int_0^3 8r \, dr = 8 \left[ \frac{r^2}{2} \right]_0^3 = 8 \cdot \frac{9}{2} = 36
Now, evaluate the outer integral:
02π36dθ=36[θ]02π=362π=72π\int_0^{2\pi} 36 \, d\theta = 36 \cdot \left[ \theta \right]_0^{2\pi} = 36 \cdot 2\pi = 72\pi
The value of the line integral is:
72π \boxed{72\pi}

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