if dy/dx = sin x cos²x and if y = 0 when x = π/2, what is the value of y when x = 0? (a) -1 (b) -1/3 (c) 0…

Explanation:

Step1: Integrate the derivative

We have $\frac{dy}{dx}=\sin x\cos^{2}x$. Let $u = \cos x$, then $du=-\sin xdx$. So $y=\int\sin x\cos^{2}x dx=-\int u^{2}du$. Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we get $y=-\frac{u^{3}}{3}+C=-\frac{\cos^{3}x}{3}+C$.

Step2: Find the constant C

Given that $y = 0$ when $x=\frac{\pi}{2}$. Substitute $x=\frac{\pi}{2}$ and $y = 0$ into $y=-\frac{\cos^{3}x}{3}+C$. Since $\cos\frac{\pi}{2}=0$, we have $0=-\frac{0^{3}}{3}+C$, so $C = 0$.

Step3: Find the value of y at x = 0

Substitute $x = 0$ into $y=-\frac{\cos^{3}x}{3}$. Since $\cos0=1$, then $y=-\frac{1^{3}}{3}=-\frac{1}{3}$.

Answer:

B. $-\frac{1}{3}$