## Task Condition

Calculate the area of the figure bounded by the graphs of the functions:

$$
y=\cos x \cdot \sin ^{2} x, y=0,\left(0 \leq x \leq \frac{\pi}{2}\right)
$$
## Solution

At points $x=0$ and $x=\frac{\pi}{2}$, the graph of the function $y=\cos x \cdot \sin ^{2} x$ intersects the x-axis, since $y(0)=y\left(\frac{\pi}{2}\right)=0$.

The sketch of the desired area is as follows:

![](https://cdn.mathpix.com/cropped/2024_05_22_aa6951b1e92815ea34b6g-13.jpg?height=1325&width=1647&top_left_y=1148&top_left_x=133)

$$
S=\int_{0}^{\pi / 2} \cos x \cdot \sin ^{2} x d x=\int_{0}^{\pi / 2} \sin ^{2} x d(\sin x)=\left.\frac{\sin ^{3} x}{3}\right|_{0} ^{\frac{\pi}{2}}=\frac{\sin ^{3} \pi / 2}{3}-\frac{\sin ^{3} 0}{3}=\frac{1^{3}}{3}-\frac{0^{3}}{3}=\frac{1}{3}
$$

\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD\%D1\%82\%D0\%B5 $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_14-7$ »

Categories: Kuznetsov's Problem Book Integrals Problem 14 | Integrals

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## Problem Kuznetsov Integrals 14-8

## Material from PlusPi