Questions: Question 1 of 6 Use the Fundamental Theorem of Calculus, Part I to find the area of the region under the graph of the function f(x)=14 cos (x) on [0, π/2]. (Use symbolic notation and fractions where needed.)

$Question 1 of 6 Use the Fundamental Theorem of Calculus, Part I to find the area of the region under the graph of the function f(x)=14 cos (x) on [0, π/2]. (Use symbolic notation and fractions where needed.)$

Transcript text: Question 1 of 6 Use the Fundamental Theorem of Calculus, Part I to find the area of the region under the graph of the function $f(x)=14 \cos (x)$ on $\left[0, \frac{\pi}{2}\right]$. (Use symbolic notation and fractions where needed.)

Solution

Solution Steps

Step 1: Define the Function

We start with the function $ f(x) = 14 \cos(x) $.

Step 2: Set Up the Integral

To find the area under the curve from $ x = 0 $ to $ x = \frac{\pi}{2} $, we set up the definite integral: \[ A = \int_{0}^{\frac{\pi}{2}} 14 \cos(x) \, dx \]

Step 3: Evaluate the Integral

Calculating the integral, we find: \[ A = 14 \left[ \sin(x) \right]_{0}^{\frac{\pi}{2}} = 14 \left( \sin\left(\frac{\pi}{2}\right) - \sin(0) \right) = 14 \left( 1 - 0 \right) = 14 \]

Final Answer

$\boxed{A = 14}$

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