Упр.1539 ГДЗ Алимов 10-11 класс (Алгебра)

2) у = х2 — 2х + 8, у = 6, х = —1, х = 3;
3) у = sin х, у = 0, х = 2пи/3, х = пи;
4) у= cos х, у = 0, х = -пи/6, х =пи/6.

1. $$S=\int_0^3\bigl(5-(4x-x^2)\bigr)\,dx=\int_0^3(5-4x+x^2)\,dx$$

   $$S=\left(5x-2x^2+\frac{x^3}{3}\right)\Bigg|_0^3=15-18+9=6$$

2. $$S=\int_{-1}^3\bigl((x^2-2x+8)-6\bigr)\,dx=\int_{-1}^3(x^2-2x+2)\,dx$$

   $$S=\left(\frac{x^3}{3}-x^2+2x\right)\Bigg|_{-1}^3$$

   $$S=\left(9-9+6\right)-\left(-\frac13-1-2\right)=6+\frac{10}{3}=\frac{28}{3}=9\frac13$$

3. $$S=\int_{2\pi/3}^{\pi}\sin x\,dx=\left(-\cos x\right)\Bigg|_{2\pi/3}^{\pi}$$

   $$S=-\cos\pi+\cos\frac{2\pi}{3}=1-\frac12=\frac12$$

4. $$S=\int_{-\pi/6}^{\pi/6}\cos x\,dx=\left(\sin x\right)\Bigg|_{-\pi/6}^{\pi/6}$$

   $$S=\sin\frac{\pi}{6}-\sin\left(-\frac{\pi}{6}\right)=\frac12+\frac12=1$$

Ответ

1) $$6$$; 2) $$9\frac13$$; 3) $$\frac12$$; 4) $$1$$.