Find the signed area produced by:[math: A=∫364cos(5x)dx](square units)Calculate your answer correct to 4 decimal places.(Note: Make sure your calculator is in radian mode.)

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BlackTom AI · Accepted Answer

The problem asks for the signed area produced by A = ∫ from 3 to 6 of 4 cos(5x) dx, with the result in square units and the calculator in radian mode.
First, compute the antiderivative of 4 cos(5x). Since ∫ cos(kx) dx = (1/k) sin(kx), we have ∫ 4 cos(5x) dx = 4 * (1/5) sin(5x) = (4/5) sin(5x).
Next, apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the bounds and subtracting: A = (4/5)[sin(5x)] from x = 3 to x = 6 = (4/5)[sin(30) − sin(15)].
Now approximate the sine values in radians: sin(30) ≈ −0.988031 and sin(15) ≈ 0.650287. Therefore, sin(30) − sin(15) ≈ −0.988031 − 0.650287 = −1.638318.
Multiply by 4/5 (which is 0.8): A ≈ −1.638318 × 0.8 = −1.310654. 
Finally, round to 4 decimal places as requested: A ≈ −1.3107. 
Note: The result is negative because the sine values lead to a net negative area over the interval when weighting by the cosine function.

UCIC 202503 EMTH118 Quiz 11

Find the signed area produced by:[math: A=∫364cos(5x)dx](square units)Calculate your answer correct to 4 decimal places.(Note: Make sure your calculator is in radian mode.)

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