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Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5 \sin x - 3 \cos x) \, dx \) - Scottish Highers Maths - Question 11 - 2023
Question 11
Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5 \sin x - 3 \cos x) \, dx \). The diagram in your answer booklet shows the graphs with equations \( y = 5 \sin x \) and \( ... show full transcript
Worked Solution & Example Answer:Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5 \sin x - 3 \cos x) \, dx \) - Scottish Highers Maths - Question 11 - 2023
Step 1
Evaluate \( \int_{\frac{\pi}{2}}^{2\pi} (5 \sin x - 3 \cos x) \, dx \)
Answer
To evaluate the integral, we first integrate the function:
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Integration:
[ \int (5 \sin x - 3 \cos x) , dx = -5 \cos x - 3 \sin x + C ]
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Substituting Limits:
Now we apply the limits from ( \frac{\pi}{2} ) to ( 2\pi ):
[ \left[ -5 \cos(2\pi) - 3 \sin(2\pi) \right] - \left[ -5 \cos\left(\frac{\pi}{2}\right) - 3 \sin\left(\frac{\pi}{2}\right) \right] ]
This simplifies to:
[ -5(1) - 3(0) - \left[ -5(0) - 3(1) \right] = -5 - (0 + 3) = -5 + 3 = -2 ]
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Final Answer:
Thus, the integral evaluates to ( -2 ).
Step 2
On the diagram in your answer booklet, shade the area represented by the integral in (a)
Answer
To shade the area represented by the integral ( \int_{\frac{\pi}{2}}^{2\pi} (5 \sin x - 3 \cos x) , dx ), locate the region between the curves ( y = 5 \sin x ) and ( y = 3 \cos x ) from ( x = \frac{\pi}{2} ) to ( x = 2\pi ).
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Identifying Boundaries:
- The lower limit is ( x = \frac{\pi}{2} ).
- The upper limit is ( x = 2\pi ).
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Shading the Area:
- Shade the area between the two curves, confirming that the area is under the curve of ( 5 \sin x ) and above ( 3 \cos x ) within these limits.