Упр.1000 ГДЗ Алимов 10-11 класс (Алгебра)

1) а = 2, b = 4, f (х) = х3;
2) а = 3,b = 4, f (х) = х2;
3) а=-2, f(x) = 1, f(x) = x2+ 1;
4) а = 0, b = 2, f(х) = х3 + 1;
5)а= пи/3, b = 2пи/3, f(x) = sinx;
6) а = -пи/6, b = 0, f (z) = cos х.

1. $$S=\int_{2}^{4}x^3\,dx=\left.\frac{x^4}{4}\right|_{2}^{4}=\frac{4^4-2^4}{4}=\frac{256-16}{4}=60.$$

2. $$S=\int_{3}^{4}x^2\,dx=\left.\frac{x^3}{3}\right|_{3}^{4}=\frac{4^3-3^3}{3}=\frac{64-27}{3}=\frac{37}{3}.$$

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

   $$=\left(\frac{1}{3}+1\right)-\left(-\frac{8}{3}-2\right)=\frac{4}{3}+\frac{14}{3}=6.$$

4. $$S=\int_{0}^{2}(x^3+1)\,dx=\left.\left(\frac{x^4}{4}+x\right)\right|_{0}^{2}=\frac{16}{4}+2=6.$$

5. $$S=\int_{\pi/3}^{2\pi/3}\sin x\,dx=\left.-\cos x\right|_{\pi/3}^{2\pi/3}=-\cos\frac{2\pi}{3}+\cos\frac{\pi}{3}$$

   $$=-\left(-\frac12\right)+\frac12=1.$$

6. $$S=\int_{-\pi/6}^{0}\cos x\,dx=\left.\sin x\right|_{-\pi/6}^{0}=\sin 0-\sin\left(-\frac{\pi}{6}\right)=0- \left(-\frac12\right)=\frac12.$$

Ответ

1) $$60$$; 2) $$\frac{37}{3}$$; 3) $$6$$; 4) $$6$$; 5) $$1$$; 6) $$\frac{1}{2}$$.