the curve x = 3 cos(y) + 12/π is graphed. what is the area bounded by the curve, the y - axis, the line y =…

Explanation:

Step1: Recall the area - formula

The area (A) between the curve (x = f(y)), the (y) - axis, and the lines (y = a) and (y = b) is given by (A=\int_{a}^{b}xdy=\int_{a}^{b}f(y)dy). Here, (f(y)=3\cos(y)+\frac{12}{\pi}), (a = \frac{\pi}{2}), and (b = 2\pi).

Step2: Integrate term - by - term

We know that (\int\left(3\cos(y)+\frac{12}{\pi}\right)dy=\int3\cos(y)dy+\int\frac{12}{\pi}dy). The integral of (\cos(y)) is (\sin(y)) and the integral of a constant (C) with respect to (y) is (Cy). So, (\int3\cos(y)dy = 3\sin(y)) and (\int\frac{12}{\pi}dy=\frac{12}{\pi}y).

Step3: Apply the fundamental theorem of calculus

(A=\left[3\sin(y)+\frac{12}{\pi}y\right]_{\frac{\pi}{2}}^{2\pi}). First, substitute (y = 2\pi) into (3\sin(y)+\frac{12}{\pi}y): (3\sin(2\pi)+\frac{12}{\pi}\times2\pi=0 + 24). Then, substitute (y=\frac{\pi}{2}) into (3\sin(y)+\frac{12}{\pi}y): (3\sin\left(\frac{\pi}{2}\right)+\frac{12}{\pi}\times\frac{\pi}{2}=3 + 6=9).

Step4: Calculate the area

(A=(24)-9 = 15).

Answer:

15