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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 \( y... 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 start by integrating the function.
-
Integrate the function:
[ \int (5\sin x - 3\cos x) , dx = -5\cos x - 3\sin x + C ]
where ( C ) is the constant of integration. -
Substitute the limits:
We compute the definite integral from ( \frac{\pi}{2} ) to ( 2\pi ):
[ F(x) = -5\cos x - 3\sin x ]
Evaluate ( F(2\pi) ):
[ F(2\pi) = -5\cos(2\pi) - 3\sin(2\pi) = -5(1) - 3(0) = -5 ]
Evaluate ( F(\frac{\pi}{2}) ):
[ F(\frac{\pi}{2}) = -5\cos(\frac{\pi}{2}) - 3\sin(\frac{\pi}{2}) = -5(0) - 3(1) = -3 ]
Now, substitute the limits into the integrated function:
[ F(2\pi) - F(\frac{\pi}{2}) = -5 - (-3) = -5 + 3 = -2 ] -
Final Answer:
[ \int_{\frac{\pi}{2}}^{2\pi} (5\sin x - 3\cos x) , dx = -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, locate the points where ( y = 5\sin x ) intersects ( y = 3\cos x ) in the range ( 0 \leq x \leq 2\pi ). The area between the curves from ( \frac{\pi}{2} ) to ( 2\pi ) should be shaded as it represents the area calculated from the integral. Make sure to highlight the area enclosed between the two curves in this interval.